Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

Abstract

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters $u=(u_1,\ldots ,u_n)$, which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters $u_1,\ldots ,u_n$. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

1 Introduction

In this paper, I answer a question asked when I presented the results of [13] and the related paper [25]. Paper [13] deals with the extension of the theory of isomonodromic deformations of the differential system (1.1), in the presence of a coalescence phenomenon involving the eigenvalues of the leading matrix $\Lambda$. These eigenvalues are the deformation parameters. The question is if we can obtain some results of [13] in terms of the Laplace transform relating system (1.1) to a Fuchsian one, such as system (1.4). The latter has simple poles at the eigenvalues of $\Lambda$, so that the coalescence of the eigenvalues will correspond to the confluence of the Fuchsian singularities. So the question is if combining integrable deformations of Fuchsian systems, confluence of singularities and Laplace transform, we can obtain the results of [13]. The positive answer is Theorem 7.1 of this paper. In order to achieve it, we extend to the case depending on deformation parameters, including their coalescence, one main result of [4, 23] concerning the existence of selected and singular vector solutions of a Pfaffian Fuchsian system associated with (1.4) (see the system (5.3)), and their connection coefficients, which will be isomonodromic. This will be obtained in Theorem 5.1 and Proposition 5.1.

In [13], the isomonodromy deformation theory of an n-dimensional differential system with Fuchsian singularity at $z=0$ and singularity of the second kind at $z=\infty$ of Poincaré rank 1

$$\begin{aligned} \frac{\mathrm{d}Y}{\mathrm{d}z}=\left( \Lambda (u)+\frac{A(u)}{z}\right) Y, \quad \quad \Lambda (u)=\hbox {diag}(u_1,\ldots ,u_n), \end{aligned}$$

(1.1)

has been considered¹. The deformation parameters $u=(u_1,\ldots ,u_n)$ vary in a polydisc where the matrix A(u) is holomorphic. One of the main results of [13] is the extension of the theory of isomonodromic deformations of (1.1) to the non-generic case when $\Lambda$ has coalescing eigenvalues but remains diagonalizable. This means that the polydisc contains a locus of coalescence points such that $u_i=u_j$ for some $1\le i\ne j\le n$. In this case, $z=\infty$ is sometimes called resonant irregular singularity. On a sufficiently small domain in the polydisc, the well-known theory of isomonodromy deformations applies and allows to define constant monodromy data. Theorem 1.1 and corollary 1.1 of [13] say that these data are well defined and constant on the whole polydisc, including the coalescence locus, if the entries of A(u) satisfy the vanishing conditions

$$\begin{aligned} (A(u))_{ij}\rightarrow 0 \hbox { when } u\hbox { tends to a coalescence point such that } u_i-u_j\rightarrow 0\hbox { at this point}. \end{aligned}$$

(1.2)

More precisely, if conditions (1.2) are satisfied, the following results (reviewed in Theorem 2.2 of Sect. 2.1) hold.

• (I) Fundamental matrix solutions in Levelt form at $z=0$ and solutions with prescribed “canonical” asymptotic behaviour in Stokes sectors at $z=\infty$ are holomorphic of u in the polydisc. Also the coefficients of the formal solution determining the asymptotics at $\infty$ are holomorphic.

• (II) Essential monodromy data, such as Stokes matrices, the central connection matrix, the formal monodromy exponent at infinity and the Levelt exponents at $z=0$ are well defined and constant on the whole polydisc, including coalescence points.

  The Stokes matrices (labelled by $\nu \in \mathbb {Z}$) satisfy the vanishing conditions

  $$\begin{aligned} (\mathbb {S}_\nu )_{ij}=(\mathbb {S}_\nu )_{ji}=0, i\ne j, \hbox { if there is a coalescence point such that } u_i=u_j. \end{aligned}$$

• (III) The constant essential monodromy data can be computed from the system “frozen” at a fixed coalescence point. In particular, if the constant diagonal entries of A are partly non-resonant (see Corollary 2.1), then there is no ambiguity in this computation, being the formal solution unique.

The results above have been established in [13] by direct analysis of system (1.1), of its Stokes phenomenon and its isomonodromic deformations.

Remark 1.1

If A(u) is holomorphic on the polydisc and (1.1) is an isomonodromic family on the polydisc minus the coalescence locus (in the sense of integrability of an associated Pfaffian system (2.14) introduced later), then (1.2) are automatically satisfied and Theorem 1.1 of [13] holds. This is not mentioned in [13]. I thank the referee for this observation. More details are in Remark 2.1.

For future use, we denote by $\lambda ^\prime _1,\dots ,\lambda ^\prime _n$ the diagonal entries of A(u), and

$$\begin{aligned} B:=\hbox {diag}(A(u))=\hbox {diag}(\lambda ^\prime _1,\dots ,\lambda ^\prime _n). \end{aligned}$$

We will see that these $\lambda ^\prime _k$ are constant in the isomonodromic case.

From another perspective, if u is fixed and $u_i\ne u_j$ for $i\ne j$, namely for a system (1.1) not depending on parameters with pairwise distinct eigenvalues of $\Lambda$, it is well known that columns of fundamental matrix solutions with prescribed asymptotics in Stokes sectors at $z=\infty$ can be obtained by Laplace-type integrals of certain selected column-vector solutions of an n-dimensional Fuchsian system of the type

$$\begin{aligned} \frac{\mathrm{d}\Psi }{ \mathrm{d}\lambda }=\sum _{k=1}^n \frac{B_k }{ \lambda -u_k}\Psi ,~~~~~B_k:=-E_k(A+I). \end{aligned}$$

(1.3)

Here, $E_k$ is the elementary matrix whose entries are zero, except for $(E_k)_{kk}=1$. These facts are studied in the seminal paper [4] in the generic case of non-integer diagonal entries $\lambda ^\prime _k$ of A. The results of [4] have been extended in [23] to the general case, when the entries $\lambda ^\prime _k$ take any complex value.

The purpose of the present paper is to introduce an isomonodromic Laplace transform relating (1.1) to an isomonodromic Fuchsian system

$$\begin{aligned} \frac{\mathrm{d}\Psi }{ \mathrm{d}\lambda }=\sum _{k=1}^n \frac{B_k(u) }{ \lambda -u_k}\Psi ,~~~~~B_k:=-E_k(A(u)+I). \end{aligned}$$

(1.4)

when $u_1,\ldots ,u_n$ vary in a polydisc containing a locus of coalescence points. More precisely, the Laplace transform will relate solutions of the integrable Pfaffian systems (2.14) and (5.3) introduced later, associated with (1.1) and (1.4), respectively. The two main goals will be:

• Theorem 5.1, which characterizes selected vector solutions and singular vector solutions of (1.4) and (5.3), so extending the results of [4] and [23] to the case depending on isomonodromic deformation parameters, including coalescing Fuchsian singularities $u_1,\ldots ,u_n$.

• Theorem 7.1, in which the Laplace transform of the vector solutions of Theorem 5.1 allows to obtain the main results (I), (II) and (III) of [13] in the presence of coalescing eigenvalues $u_1,\ldots ,u_n$ of $\Lambda (u)$.

In details.

• In Proposition 3.1 we establish the equivalence between the “strong” isomonodromic deformations (non-normalized Schlesinger deformations) of (1.4) and the ”strong” isomonodromic deformations of (1.1).

• Then, we study isomonodromy deformations of (1.4) when u varies in a polydisc containing a coalescence locus. Theorem 5.1, provides selected and singular vector solutions, which are the isomonodromic analogue of solutions introduced in [4, 23], respectively, denoted by $\vec {\Psi }_k(\lambda ,u~ |\nu )$ and $\vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu )$, $k=1,\ldots ,n$, the latter being singular at $\lambda =u_k$. As will be explained later, $\nu \in \mathbb {Z}$ labels the directions of branch cuts in the punctured $\lambda$-plane at the poles $u_1,\ldots ,u_n$. These solutions allow to introduce connection coefficients $c_{jk}^{(\nu )}$, defined by

  $$\begin{aligned} \vec {\Psi }_k(\lambda ,u~ |\nu )=\vec {\Psi }^{(sing)}_j(\lambda ,u~ |\nu ) c_{jk}^{(\nu )} +\hbox {holomorphic part at } \lambda =u_j, \quad \forall ~ j\ne k. \end{aligned}$$

  The above is the deformation parameters dependent analogue of the definition of connection coefficients in [23].

• In Proposition 5.1, we prove that the $c_{jk}^{(\nu )}$ are isomonodromic connection coefficients, namely independent of u. When there is a coalescence $u_j=u_k$ in the polydisc, they satisfy

  $$\begin{aligned} c_{jk}^{(\nu )}=0,\quad j\ne k. \end{aligned}$$

• In Theorem 7.1, the Laplace transform of the vectors $\vec {\Psi }_k(\lambda ,u~ |\nu )$ or $\vec {\Psi }_k^{(sing)}(\lambda ,u~ |\nu )$ yields the columns of the isomonodromic fundamental matrix solutions $Y_\nu (z,u)$ of (1.1), labelled by $\nu \in \mathbb {Z}$, uniquely determined by a prescribed asymptotic behaviour in certain u-independent sectors $\widehat{\mathcal {S}}_\nu$, of central opening angle greater than $\pi$. The analytic properties for the matrices $Y_\nu (z,u)$ will be proved, so re-obtaining the result (I) above. In order to describe the Stokes phenomenon, only three solutions $Y_\nu (z,u)$, $Y_{\nu +\mu }(z,u)$ and $Y_{\nu +2\mu }(z,u)$ suffice. The labelling will be explained later. The Stokes matrices $\mathbb {S}_{\nu +k\mu }$, $k=0,1$, defined by a relation $Y_{\nu +(k+1)\mu }=Y_{\nu +k\mu } \mathbb {S}_{\nu +k\mu }$, will be expressed in terms of the coefficients $c_{jk}^{(\nu )}$ in formula (7.9). This extends to the isomonodromic case, including coalescences, an analogous expression appearing in [4, 23] and implies the results in (II) above.

• In Sect. 8, we re-obtain the result (III), that system (1.1), “frozen” by fixing u equal to the most coalescence point $u^c$ in the polydisc (see Sect. 2.1 for $u^c$), admits a unique formal solution if and only if the (constant) diagonal entries $\lambda _j^\prime$ of A satisfy $\lambda _i^\prime -\lambda _j^\prime \not \in \mathbb {Z}\backslash \{0\}$ for every $i\ne j$ such that $u^c_i=u^c_j$. In this case we prove that the selected vector solutions of the Fuchsian system (1.4) at $u=u^c$, needed to perform the Laplace transforms, are uniquely determined. On the other hand, if some $\lambda _i^\prime -\lambda _j^\prime \in \mathbb {Z}\backslash \{0\}$ corresponding to $u_i^c=u_j^c$, then there is a family of solutions of the Fuchsian system (1.4) at a coalescence point, depending on a finite number of parameters: this facts is responsible, through the Laplace transform, of the existence of a family of formal solutions at the coalescence point.

In [19, 20], B. Dubrovin related system (1.1) to an isomonodromic system of type (1.4), in the specific case when they, respectively, yield the flat sections of the deformed connection of a semisimple Dubrovin–Frobenius manifold and the flat sections of the intersection form (extended Gauss-Manin system). In [19, 20], the solutions of (1.1) are expressed by Laplace transform of the isomonodromic system (1.4), but the eigenvalues $u_1,\ldots ,u_n$ are assumed to be pairwise distinct in a sufficiently small domain (analogous to the polydisc $\mathbb {D}(u^0)$ to be introduced later). Moreover, A is skew-symmetric, so its diagonal elements are zero (A is denoted by V and $\Lambda$ by U in [19, 20]). By a Coxeter-type identity, the entries of the monodromy matrices for the selected solutions of (1.4) (which are part of the monodromy of the Dubrovin–Frobenius manifold) are expressed in terms of the entries of the Stokes matrices. See also [21, 61].

In proposition 2.5.1 of [22], the authors prove (I) when system (1.1) is associated with a Dubrovin–Frobenius manifold with semisimple coalescence points, and A is skew-symmetric (in [22] the irregular singularity is at $z=0$). Their proof contains the core idea that the analytic properties of a solution Y(z, u) in (I) are obtainable, by a Laplace transform, from the analytic properties of a fundamental matrix solution $\Psi (\lambda ,u)$ of the Fuchsian Pfaffian system associated with (1.4) (see their Lemma 2.5.3). The latter is a particular case of the Pfaffian systems studied in [63]. On the other hand, the analysis of selected and singular vector solutions of the Fuchsian Pfaffian system, required in our paper to cover all possible cases (all possible A), is not necessary in [22], due to the skew-symmetry of A, and the specific form of their Pfaffian system (see their equation (2.5.2); their discussion is equivalent our case $\lambda ^\prime _j=-1$ for all $j=1,\ldots ,n$). Moreover, points (II) and (III) are not discussed in [22] by means of the Laplace transform.

In the present paper, by an isomonodromic Laplace transform, we prove (I), (II) and (III), and at the same time we generalize the results of [4, 23] to the isomonodromic case with coalescences, with no assumptions on the eigenvalues and the diagonal entries of A. This analytic construction, to the best of our knowledge, cannot be found in the literature.

The approach of the present paper may also be used to extend the results of [19, 20] described above, relating the deformed flat connection and the intersection form, namely Stokes matrices and monodromy group of the Dubrovin–Frobenius manifold, in case of semisimple coalescent Frobenius structures studied in [10, 14, 15, 17].

For further comments and reference on the use of the Laplace transform, the confluence of singularities and related topics, see the introduction of [23] and [9, 29, 36,37,38,39, 44, 49, 56,57,58,59].

Stickily related to ours are the important results of [52]. In [13] (and in the present paper by Laplace transform), we have answered the question if the integrable deformation (2.14) of system (1.1) extends from a polydisc (or a small open set) not containing coalescence points to a wider domain intersecting (a stratum of) the coalescence locus, and we have characterized the monodromy data. The converse question is answered in [52], namely if an integrable deformation (2.14) of $(\Lambda (u^c) +A(u^c)/z)\mathrm{d}z$ exists and is unique, having formal normal form $d(z\Lambda (u))+B/z ~\mathrm{d}z$, where B is the diagonal of $A(u^c)$. More broadly, the question of [52] is the existence and uniqueness of integrable deformations of meromorphic connections on $\mathbb {P}^1$ with irregular singularity, when a prefixed restriction is given at a single point $t_o$ in the space of deformation parameters T, allowed to be a degenerate point, namely a coalescence point in our case (in [52], deformation parameters are called $t\in T$). One asks if a connection $\omega (z,t_0)$ given at $t_o\in T$ can be deformed to $\omega (z,t)$, and if this deformation is unique.² Concerning uniqueness, for a fixed normal form $\omega _0(z,t)$, the problem is to classify isomorphism pairs $(\omega ,G)$ consisting of an integrable connection $\omega (z,t)$ (with poles in $T\times \{z=0\}$, being $z=0$ used in [52], while $z=\infty$ is used in our works) and a formal gauge transformation G(z, t) (formal in z but holomorphic in t), transforming $\omega (z,t)$ to $\omega _0(z,t)$. In a general context, a uniqueness theorem is proved in [60]: two pairs are isomorphic (meaning that the composition of a gauge of one pair with the inverse gauge of the other pair is convergent w.r.t. z) if and only if their restriction to any specific value $t_o$ are isomorphic. Thus, the t-extension of a pair in a neighbourhood of $t_0$ is unique up to isomorphism. The proof in [60] makes use of the results of Kedlaya [34, 35] and Mochizuki [45,46,47,48], which allows to blow up $T\times \{0\}$, and of the higher-dimensional asymptotic analysis in poli-sectors for the formal gauge transformations, that is Majima’s asymptotic analysis [40] for Pfaffian systems with irregular singularities. In [52], the uniqueness result is proved for a restricted class of integrable connections, in which our (2.14) is contained (with irregular singularity at $z=0$ instead of $\infty$). So, given a block-diagonal normal form $\omega _0(z,t)$ and a pair consisting of $\omega (t_o,z)$ and a formal gauge $G(t_o,z)$, it is proved that the pair can be deformed (existence) in a unique way (uniqueness) to $\omega (z,t), G(z,t)$, such that $G[\omega ]=\omega _0$. The strategy is to use a sequence of Kedlaya–Mochizuki blow-ups to obtain a good normal form (see also [50, 51]). Then, Majima results on asymptotic analysis can be used and adapted. In our specific case, theorem 4.9 of [52] means the existence and uniqueness of the integrable deformation (2.14) of $(\Lambda (u^c) +A(u^c)/z)\mathrm{d}z$, formally equivalent to $d(z\Lambda (u))+B/z \mathrm{d}z$. These facts generalize results of Malgrange [41, 42] for irregular singularities to the case of coalescence points.

Theorem 4.9, obtained in [52] in geometric terms, has been successively proved in [11] by analytic methods. In [11], the integrable deformation is obtained from prefixed monodromy data at a coalescence point, using the analytic $L^p$ theory a Riemann–Hilber boundary value problems. Both authors of [52] and [11] apply their results to semisimple Dubrovin–Frobenius manifolds. In particular, [11] proves that any formal semisimple Frobenius manifold is the completion of a pointed germ of an analytic Dubrovin–Frobenius manifold. The result is extended to F-manifolds in the recent work [12].

A geometric formulation of the Laplace transform we have used here, together with a synthetic proof of part of Theorem 1.1 of [13], is the object of the recent work [53].

2 Review of background material

This section contains known material to motivate and understand our paper. For X a topological space, we denote by $\mathcal {R}(X)$ its universal covering. For $\alpha <\beta \in \mathbb {R}$, a sector is written as

$$\begin{aligned} S(\alpha , \beta ):=\{z\in \mathcal {R}(\mathbb {C}\backslash \{0\}) \hbox { such that } \alpha<\arg z <\beta \} . \end{aligned}$$

2.1 Background 1: isomonodromy deformations of (1.1) with coalescing eigenvalues

We review some results of [13, 25] (see also [16, 24, 26]). Consider a differential system (1.1) with an $n\times n$ with matrix coefficient A(u) holomorphic in a polydisc

$$\begin{aligned} \mathbb {D}(u^c):=\{u\in \mathbb {C}^n\hbox { such that } \max _{1\le j\le n} |u_j-u_j^c|\le \epsilon _0\},\quad \epsilon _0>0, \end{aligned}$$

(2.1)

centered at a coalescence point $u^c=(u_1^c,\ldots ,u_n^c)$, so called because

$$\begin{aligned} u_i^c=u_j^c\quad \hbox { for some } i\ne j. \end{aligned}$$

The eigenvalues of $\Lambda (u)$ coalesce at $u^c$ and also along the following coalescence locus

$$\begin{aligned} \Delta :=\mathbb {D}(u^c)\cap \Bigl (\bigcup _{i\ne j}\{u_i-u_j=0\}\Bigr ), \end{aligned}$$

We assume that $\mathbb {D}(u^c)$ is sufficiently small so that $u^c$ is the most coalescent point. Namely, if $u_j^c\ne u_k^c$ for some $j\ne k$, then $u_j\ne u_k$ for all $u\in \mathbb {D}(u^c)$. A more precise characterization of the radius $\epsilon _0$ of the polydisc will be given in Sect. 5. For $u^0\in \mathbb {D}(u^c)\backslash \Delta$, let

$$\begin{aligned} \mathbb {D}(u^0)\subset ( \mathbb {D}(u^c)\backslash \Delta ) \end{aligned}$$

be a (smaller) polydisc centered at $u^0$, not containing coalescence points.

2.1.1 Deformations in $\mathbb {D}(u^0)$

If $\mathbb {D}(u^0)$ is sufficiently small, the isomonodromic theory of Jimbo, Miwa and Ueno [33] assures that the essential monodromy data of (1.1) (see Definition 2.1) are constant over $\mathbb {D}(u^0)$ and can be computed fixing $u=u^0$.

In order to give fundamental solutions with “canonical” form at $z=\infty$, in $\mathcal {R}(\mathbb {C}\backslash \{0\})$ we introduce the Stokes rays of $\Lambda (u^0)$, defined by

$$\begin{aligned} \mathfrak {R}((u_j^0-u_k^0)z)=0, \quad \mathfrak {I}((u_j^0-u_k^0)z) <0, \quad 1\le j\ne k \le n. \end{aligned}$$

Let

$$\begin{aligned} \arg z =\tau ^{(0)} \end{aligned}$$

(2.2)

be a direction which does not coincide with any of the Stokes rays of $\Lambda (u^0)$, called admissible at $u^0$. Each sector of amplitude $\pi$, whose boundaries are not Stokes rays of $\Lambda (u^0)$, contains a certain number $\mu ^{(0)}\ge 1$ of Stokes rays of $\Lambda (u^0)$, with angular directions

$$\begin{aligned} \arg z ~=~ \tau _{0},\tau _{1},\ldots ,\tau _{\mu ^{(0)}-1},\quad \hbox { with } \quad \tau _{0},<\tau _{1}<\cdots <\tau _{\mu ^{(0)}-1} \end{aligned}$$

that we decide to label from 0 to $\mu ^{(0)}-1$. They are basic rays, since they generate all the Stokes rays in $\mathcal {R}(\mathbb {C}\backslash \{0\})$ associated with $\Lambda (u^0)$ by the formula

$$\begin{aligned} \arg z=\tau _{\nu }:=\tau _{\nu _0} +k\pi ,\quad \quad \nu _0\in \{ 0,\ldots , \mu ^{(0)}-1\}, \quad \quad \nu =\nu _0+k\mu ^{(0)},\quad k\in \mathbb {Z}. \end{aligned}$$

The choice to label a specific Stokes ray with 0, as $\tau _0$ above, is arbitrary, and it induces the labelling $\nu \in \mathbb {Z}$ for all other rays. Suppose the labelling has been chosen. Then, for some $\nu \in \mathbb {Z}$, we have

$$\begin{aligned} \tau _\nu<\tau ^{(0)}<\tau _{\nu +1}. \end{aligned}$$

(2.3)

Equivalently, given $\tau ^{(0)}$, one can choose a $\nu$ and decide to call $\tau _\nu$ and $\tau _{\nu +1}$ the Stokes rays satisfying (2.3). This induces the labelling of all other rays (notice that $\mu ^{(0)}$ is not a choice!).

Similarly, we consider the Stokes rays $\mathfrak {R}((u_j-u_k)z)=0$, $\mathfrak {I}((u_j-u_k)z) <0$ of $\Lambda (u)$. If $\mathbb {D}(u^0)$ is sufficiently small, when u varies the Stokes rays of $\Lambda (u)$ rotate without crossing $\arg z=\tau ^{(0)}$ mod $\pi$. For $k\in \mathbb {Z}$, we take the sector $S\bigl (\tau ^{(0)}+(k-1)\pi ,\tau ^{(0)}+k\pi \bigr )$ and extend it in angular amplitude up to the nearest Stokes rays of $\Lambda (u)$ outside. The resulting (open) sector will be denoted by $\mathcal {S}_{\nu +k\mu ^{(0)}}(u)$, and we define

$$\begin{aligned} \mathcal {S}_{\nu +k\mu ^{(0)}}( \mathbb {D}(u^0)):= \bigcap _{u\in \mathbb {D}(u^0)}\mathcal {S}_{\nu +k\mu ^{(0)}}(u). \end{aligned}$$

The reason for the labelling is that $S\bigl (\tau ^{(0)}+(k-1)\pi ,\tau ^{(0)}+k\pi \bigr )\subset S(\tau _{\nu +k\mu ^{(0)}}-\pi ,~\tau _{\nu +k\mu ^{(0)}+1})$ and consequently

$$\begin{aligned} {\mathcal {S}}_{\nu +k\mu ^{(0)}}(\mathbb {D}(u^0))\subset S(\tau _{\nu +k\mu ^{(0)}}-\pi ,~\tau _{\nu +k\mu ^{(0)}+1})\equiv S(\tau _{[\nu +k\mu ^{(0)}]-\mu ^{(0)}},~\tau _{[\nu +k\mu ^{(0)}]+1}). \end{aligned}$$

By construction, $\mathcal {S}_\nu ( \mathbb {D}(u^0))$ has central angular opening greater than $\pi$. See Fig. 1.

Fig. 1

Successive sectors $\mathcal {S}_\nu (\mathbb {D}(u^0))$ and $\mathcal {S}_{\nu +\mu }(\mathbb {D}(u^0))$. Their intersection (in the right part of the figure) does not contain Stokes rays. It contains the admissible direction $\arg z=\tau ^{(0)}$

Proposition 2.1

(Sibuya [30, 54, 55]; see also [13, 25, 33]). Let $\mathbb {D}(u^0)$, not containing coalescence points, be sufficiently small so that the Stokes rays of $\Lambda (u)$ do not cross³ the admissible rays $\arg z=\tau ^{(0)}+h\pi$, $h\in \mathbb {Z}$, as u varies in $\mathbb {D}(u^0)$. System (1.1) has a unique formal solution

$$\begin{aligned} Y_F(z,u)=F(z,u) z^{B(u)} \exp \{z\Lambda (u)\},\quad \quad B(u):=\hbox {diag}(A_{11}(u),\ldots ,A_{nn}(u)), \end{aligned}$$

(2.4)

where

$$\begin{aligned} F(z,u)=I+\sum _{k=1}^\infty F_k(u) z^{-k} \end{aligned}$$

(2.5)

is a formal series, with holomorphic matrix coefficients $F_k(u)$.For every $\nu \in \mathbb {Z}$, there exist unique fundamental matrix solutions

$$\begin{aligned} Y_\nu (z,u)=\widehat{Y}_\nu (z,u)z^{B(u)} \exp \{z\Lambda (u)\} \end{aligned}$$

(2.6)

of (1.1), holomorphic on $\mathcal {R}\bigl (\mathbb {C}\backslash \{0\}\times \mathbb {D}(u^0)\bigr )\equiv \mathcal {R}(\mathbb {C}\backslash \{0\})\times \mathbb {D}(u^0)$, such that uniformly in $u\in \mathbb {D}(u^0)$ the following asymptotic behaviour holds

$$\begin{aligned} \widehat{Y}_\nu (z,u)\sim F(z,u)\quad \hbox { for } z\rightarrow \infty \hbox { in } \mathcal {S}_\nu ( \mathbb {D}(u^0)). \end{aligned}$$

(2.7)

The coefficients $F_k$ are computed recursively [13, 62]

$$\begin{aligned}&(F_1)_{i j}=\frac{A_{ij}}{u_j-u_i},~i\ne j, \quad \quad \quad (F_1)_{ii}=-\sum _{j\ne i}A_{ij}(F_1)_{ji}, \end{aligned}$$

(2.8)

$$\begin{aligned}&(F_k)_{i j}=\frac{1}{u_j-u_i}\left\{ \Bigl (A_{i i}-A_{jj}+k-1 \Bigr )(F_{k-1})_{i j} +\sum _{p\ne i}A_{i p}(F_{k-1})_{p j}\right\} , \quad i\ne j; \end{aligned}$$

(2.9)

$$\begin{aligned}&k(F_{k})_{i i}=-\sum _{j\ne i }A_{i j}(F_{k})_{ji}. \end{aligned}$$

(2.10)

Holomorphic Stokes matrices $\mathbb {S}_\nu (u)$, $\nu \in \mathbb {Z}$, are the connection matrices defined by

$$\begin{aligned} Y_{\nu +\mu ^{(0)}}(z,u)=Y_\nu (z,u) \mathbb {S}_\nu (u),\quad \quad z\in \mathcal {S}_\nu (\mathbb {D}(u^0))\cap \mathcal {S}_{\nu +\mu ^{(0)}}(\mathbb {D}(u^0)). \end{aligned}$$

(2.11)

Notice that $\mathcal {S}_\nu (\mathbb {D}(u^0))\cap \mathcal {S}_{\nu +\mu ^{(0)}}(\mathbb {D}(u^0))$ does not contain Stokes rays of $\Lambda (u)$, for every $u\in \mathbb {D}(u^0)$.

At every fixed $u\in \mathbb {D}(u^0)$, system (1.1) admits a fundamental matrix solution in Levelt form

$$\begin{aligned} Y^{(0)}(z,u)=G^{(0)}(u)\Bigl (I+\sum _{j=1}^\infty \Psi _j(u)z^j\Bigr )z^Dz^L, \end{aligned}$$

(2.12)

where the series is convergent absolutely in every ball $|z|<N$, for every $N>0$. Here, D is diagonal with integer entries (called valuations), L has eigenvalues with real part lying in [0, 1), and $D+\lim _{z\rightarrow 0}z^D L z^{-D}$ is a Jordan form of A. A central connection matrix $C_\nu (u)$ is defined by

$$\begin{aligned} Y_\nu (z,u)=Y^{(0)}(z,u)C_\nu (u). \end{aligned}$$

(2.13)

A pair of Stokes matrices $\mathbb {S}_\nu$, $\mathbb {S}_{\nu +\mu ^{(0)}}$, together with B, $C_\nu$ and L are sufficient to calculate all the other $\mathbb {S}_{\nu ^\prime }$ and $C_{\nu ^\prime }$, for all $\nu ^\prime \in \mathbb {Z}$ (see [1, 13]). The monodromy matrices at $z=0$ are

$$\begin{aligned} M:=e^{2\pi i L} \quad \hbox { and } \quad e^{2\pi i B} (\mathbb {S}_\nu \mathbb {S}_{\nu +\mu ^{(0)}})^{-1}=C_\nu ^{-1}MC_\nu \end{aligned}$$

for $Y^{(0)}$ and $Y_\nu$, respectively. Hence, it makes sense to define strong isomonodromy deformations, as follows.

Definition 2.1

Fixed a $\nu \in \mathbb {Z}$, we call essential monodromy data the matrices

$$\begin{aligned} \mathbb {S}_\nu , \quad \mathbb {S}_{\nu +\mu ^{(0)}}, \quad B, \quad C_\nu , \quad L, \quad D. \end{aligned}$$

The deformation u is strongly isomonodromic on $\mathbb {D}(u^0)$, if the essential monodromy data are constant on $\mathbb {D}(u^0)$.

We introduced the terminology strong in [25], to mean that all the essential monodromy data are constant, contrary to the case of weak isomonodromic deformations, which only preserve monodromy matrices of a certain fundamental matrix solution. For a deformation to be weakly isomonodromic it is necessary and sufficient that (1.1) is the z-component of a certain Pfaffian system $dY=\omega (z,u) Y$, Frobenius integrable (i.e. $\mathrm{d}\omega =\omega \wedge \omega$). If $\omega$ is of very specific form, the deformation becomes strongly isomonodromic, according to the following

Theorem 2.1

System (1.1) is strongly isomonodromic in $\mathbb {D}(u^0)$ if and only $Y_\nu (z,u)$, for every $\nu$, and $Y^{(0)}(z,u)$, satisfy the Frobenius integrable Pfaffian system

$$\begin{aligned} dY=\omega (z,u) Y,\quad \quad \omega (z,u)=\left( \Lambda (u)+\frac{A(u)}{z}\right) \mathrm{d}z+\sum _{k=1}^n \omega _k(z,u) \mathrm{d}u_k, \end{aligned}$$

(2.14)

with the matrix coefficients (here $F_1$ is in (2.8))

$$\begin{aligned} \omega _k(z,u)= zE_k+ \omega _k(u),\quad \quad \omega _k(u)= [F_1(u),E_k]. \end{aligned}$$

(2.15)

Equivalently, (1.1) is strongly isomonodromic if and only if⁴A satisfies

$$\begin{aligned} dA=\sum _{j=1}^n \Bigl [\omega _k(u) ,A\Bigr ]\mathrm{d}u_k . \end{aligned}$$

(2.16)

If the deformation is strongly isomonodromic, then $Y^{(0)}(z,u)$ in (2.12) is holomorphic on $\mathcal {R}(\mathbb {C}\backslash \{0\})\times \mathbb {D}(u^0)$, with holomorphic matrix coefficients $\Psi _j(u)$, and the series is convergent uniformly w.r.t. $u\in \mathbb {D}(u^0)$. Moreover, $G^{(0)}(u)$ is a holomorphic fundamental solution of the integrable Pfaffian system

$$\begin{aligned} dG=\Bigl (\sum _{j=1}^n\omega _k(u)\mathrm{d}u_k\Bigr )G, \end{aligned}$$

(2.17)

and A(u) is holomorphically similar to the Jordan form $J=G^{(0)}(u)^{-1}A(u)G^{(0)}(u)$.

The above theorem is analogous to the characterization of isomonodromic deformations in [33], but includes also possible resonances in A (see [13] and Appendix B of [25]). Notice that $\omega (z,u)$ in (2.14)–(2.15) has components

$$\begin{aligned} \omega _k(u)= \left( \frac{A_{ij}(\delta _{ik}-\delta _{jk})}{u_i-u_j} \right) _{i,j=1}^n =\left( \begin{array}{ccccc} 0 &{} 0&{}\frac{ -A_{1k}}{u_1-u_k} &{} 0&{} 0 \\ 0 &{}0 &{}\vdots &{} 0&{} 0 \\ \frac{ A_{k1}}{u_k-u_1} &{} \cdots &{}0 &{} \cdots &{}\frac{ A_{kn}}{u_k-u_n} \\ 0 &{}0 &{}\vdots &{}0 &{} 0 \\ 0 &{}0 &{}\frac{ -A_{nk}}{u_n-u_k} &{} 0&{} 0 \end{array} \right) \end{aligned}$$

(2.18)

Notice that $B=\hbox {diag}(A(u))=\hbox {diag}(\lambda ^\prime _1,\ldots ,\lambda ^\prime _n)$ is constant because (2.16) and (2.18) imply

$$\begin{aligned} \frac{\partial A_{ii}}{\partial u_j}=0, \quad \forall i,j=1,\ldots ,n. \end{aligned}$$

2.1.2 Deformations in $\mathbb {D}(u^c)$ with coalescences

When the polydisc contains a coalescence locus $\Delta$, the analysis presents problematic issues.

• A fundamental matrix solution Y(z, u) holomorphic on $\mathcal {R}\bigl ((\mathbb {C}\backslash \{0\})\times (\mathbb {D}(u^c))\backslash \Delta )\bigr )$ may be singular at $\Delta$, namely the limit for $u\rightarrow u^*\in \Delta$ along any direction may diverge, and $\Delta$ is in general a branching locus [43].

• The monodromy data associated with a fundamental matrix solution $\mathring{Y}(z)$ of

  $$\begin{aligned} \frac{\mathrm{d}Y}{\mathrm{d}z}=\left( \Lambda (u^c)+\frac{A(u^c)}{z}\right) Y, \end{aligned}$$

  (2.19)

  differ from those of any fundamental solution Y(z, u) of (1.1) at $u\not \in \Delta$ ([2, 3, 13]).

In $\mathcal {R}(\mathbb {C}\backslash \{0\})$, we introduce the Stokes rays of $\Lambda (u^c)$

$$\begin{aligned} \mathfrak {R}((u_i^c-u_k^c)z)=0, \quad \mathfrak {I}((u_i^c-u_k^c)z) <0, \quad u_i\ne u_k, \end{aligned}$$

and an admissible direction at $u^c$

$$\begin{aligned} \arg z=\tau , \end{aligned}$$

(2.20)

such that none of the Stokes rays at $u=u^c$ take this direction. Notice that $\tau$ is associated with $u^c$, differently from $\tau ^{(0)}$ of Sect. 2.1.1. We choose $\mu$ basic Stokes rays of $\Lambda (u^c)$. These are all and the only Stokes rays lying in a sector of amplitude $\pi$, whose boundaries are not Stokes rays of $\Lambda (u^c)$. Notice that $\mu$ is different from $\mu ^{(0)}$ used in Sect. 2.1.1. We label their directions $\arg (z)$ as follows:

$$\begin{aligned} \tau _0<\tau _1<\cdots <\tau _{\mu -1}. \end{aligned}$$

The directions of all the other Stokes rays of $\Lambda (u^c)$ in $\mathcal {R}(\mathbb {C}\backslash \{0\})$ are consequently labelled by an integer $\nu \in \mathbb {Z}$

$$\begin{aligned} \arg z=\tau _\nu :=\tau _{\nu _0}+k\pi ,\quad \hbox { with } \nu _0\in \{0,\ldots ,\mu -1\}\hbox { and } \nu :=\nu _0+k\mu . \end{aligned}$$

(2.21)

They satisfy $\tau _{\nu }<\tau _{\nu +1}$.

Analogously, at any other $u\in \mathbb {D}(u^c)$, we define Stokes rays $\mathfrak {R}((u_i-u_j)z)=0$, $\mathfrak {I}((u_i-u_j)z)<0$ of $\Lambda (u)$. They behave differently from the case of $\mathbb {D}(u^0)$. Indeed, if u varies in $\mathbb {D}(u^c)$, some Stokes rays cross the admissible directions $\arg z= \tau$ mod $\pi$, as follows. Let i, j, k be such that $u_i^c=u_j^c\ne u_k^c$. Then, as u moves away from $u^c$, a Stokes ray of $\Lambda (u^c)$ characterized by $\mathfrak {R}((u_i^c-u_k^c)z)=0$ generates three rays. Two of them are $\mathfrak {R}((u_i-u_k)z)=0$ and $\mathfrak {R}((u_j-u_k)z)=0$. If $\mathbb {D}(u^c)$ is sufficiently small (as in (5.1)), they do not cross $\arg z= \tau$ mod $\pi$ as u varies in $\mathbb {D}(u^c)$. The third ray is $\mathfrak {R}((u_i-u_j)z)=0$. When u varies in $\mathbb {D}(u^c)$ making a complete loop $(u_i-u_j)\mapsto (u_i-u_j)e^{2\pi i}$ around the locus $\{u\in \mathbb {D}(u^c)~|~u_i-u_j=0\}\subset \Delta$, the third ray crosses $\arg z= \tau$ mod $2\pi$ and $\arg z= \tau -\pi$ mod $2\pi$. This identifies a crossing locus $X(\tau )\subset \mathbb {D}(u^c)$ of points u such that there exists a Stokes ray of $\Lambda (u)$ (so infinitely many in $\mathcal {R}(\mathbb {C}\backslash \{0\})$) with direction $\tau$ mod $\pi$.

Proposition 2.2

([13]). Each connected component of $\mathbb {D}(u^c)\backslash (\Delta \cup X(\tau ))$ is simply connected and homeomorphic to a ball, so it is a topological cell.

Thus, the choice of $\tau$ induces a cell decomposition of $\mathbb {D}(u^c)$. Each cell is called $\varvec{\tau }$ -cell. If u varies in the interior of a $\tau$-cell, no Stokes rays cross the admissible directions $\arg z= \tau +h\pi$, $h\in \mathbb {Z}$, but if u varies in the whole $\mathbb {D}(u^c)$, then $X(\tau )$ is crossed, and thus Proposition 2.1 does not hold.

To overcome this difficulty, we first take a point $u^0$ in a $\tau$-cell, and consider a polydisc $\mathbb {D}(u^0)$ contained in the $\tau$-cell, satisfying the assumptions of Sect. 2.1.1. Accordingly, we can define as before the sectors ${\mathcal {S}}_{\nu +k\mu }(u)$ of angular amplitude greater than $\pi$, and

$$\begin{aligned} {\mathcal {S}}_{\nu +k\mu }(\mathbb {D}(u^0))= \bigcap _{u\in \mathbb {D}(u^0)}\mathcal {S}_{\nu +k\mu }(u)\subset \{\tau _{\nu +k\mu }-\pi<\arg z <\tau _{\nu +k\mu +1}\}. \end{aligned}$$

Notice that here we are using $\tau$ and $\mu$ in place of $\tau ^{(0)}$ and $\mu ^{(0)}$. With the above sectors, monodromy data in (2.11)–(2.13) can be defined in $\mathbb {D}(u^0)$.

Since A(u) is holomorphic in $\mathbb {D}(u^0)$, then $\omega _k(z,u)$ is holomorphic on $\mathbb {D}(u^c)\backslash \Delta$. Thus, the fundamental matrix solutions $Y_\nu (z,u)$, $Y^{(0)}(z,u)$ of Sect. 2.1.1 extend analytically on $\mathcal {R}\bigl ((\mathbb {C}\backslash \{0\})\times (\mathbb {D}(u^c))\backslash \Delta )\bigr )\ne \mathcal {R}(\mathbb {C}_z\backslash \{0\})\times (\mathbb {D}(u^c))\backslash \Delta )$, and $\Delta$ may be a branching locus for them.

Proposition 2.3

([13]). $\omega (z,u)$ in (2.15) and (2.18) is holomorphic on the whole $\mathbb {D}(u^c)$ if and only if

$$\begin{aligned} A_{ij}(u)=\mathcal {O}(u_i-u_j)\rightarrow 0 \quad \hbox { whenever } (u_i-u_j)\rightarrow 0\hbox { for } u\hbox { approaching } \Delta . \end{aligned}$$

A(u) is holomorphically similar on $\mathbb {D}(u^c)$ to a Jordan form J if and only if the above vanishing conditions hold. Similarity is realized by a fundamental matrix solution of (2.17), which exists holomorphic on the whole $\mathbb {D}(u^c)$.

The extension of the theory of isomonodromy deformations on the whole $\mathbb {D}(u^c)$ is given in [13] by the following theorem, which is a detailed exposition of the points (I) and (II) of Introduction, while point (III) is expressed by Corollary 2.1.

Theorem 2.2

([13]). Let A(u) be holomorphic on $\mathbb {D}(u^c)$. Assume that system (1.1) is strongly isomonodromic on $\mathbb {D}(u^0)$ contained in a $\tau$-cell of $\mathbb {D}(u^c)$, so that Theorem 2.1 holds. Moreover, assume that A satisfies the vanishing conditions

$$\begin{aligned} A_{ij}(u)=\mathcal {O}(u_i-u_j)\rightarrow 0 \quad \hbox { whenever } (u_i-u_j)\rightarrow 0\hbox { for } u\hbox { approaching } \Delta . \end{aligned}$$

(2.22)

Then, the following statements hold.

Part I.

• (I,1) $Y^{(0)}(z,u)$ and the $Y_\nu (z,u)$, $\nu \in \mathbb {Z}$ admit analytic continuation as holomorphic functions on $\mathcal {R}(\mathbb {C}\backslash \{0\})\times \mathbb {D}(u^c)$. The coalescence locus $\Delta$ is neither a singularity locus nor a branching locus.

• (I,2) The coefficients $F_k(u)$ of $Y_F(z,u)$, given in (2.8)–(2.9)–(2.10), are holomorphic of $u\in \mathbb {D}(u^c)$.

• (I,3) The fundamental matrix solutions $Y_\nu (z,u)$ have asymptotics $Y_\nu (z,u) \sim Y_F(z,u)$ uniformly in $u\in \mathbb {D}(u^c)$, for $z\rightarrow \infty$ in a wide sector $\widehat{\mathcal {S}}_{\nu }$ containing $\mathcal {S}_{\nu }(\mathbb {D}(u^0))$, to be defined later in (7.3).

Part II.

• (II,1) the essential monodromy data $\mathbb {S}_\nu$, $\mathbb {S}_{\nu +\mu }$, $B=\hbox {diag}(A(u^c))$, $C_\nu$, L, D, initially defined on $\mathbb {D}(u^0)$ by relations (2.11)–(2.13), are well defined and constant on the whole $\mathbb {D}(u^c)$. They satisfy

  $$\begin{aligned} \mathbb {S}_\nu =\mathring{\mathbb {S}}_\nu ,\quad \mathbb {S}_{\nu +\mu }=\mathring{\mathbb {S}}_{\nu +\mu }, \quad L=\mathring{L}, \quad C_\nu =\mathring{C}_\nu , \quad D=\mathring{D}, \end{aligned}$$

  where

• (II,2) $\mathring{\mathbb {S}}_\nu$, $\mathring{\mathbb {S}}_{\nu +\mu }$ are the Stokes matrices of fundamental solutions $\mathring{Y}_\nu (z)$, $\mathring{Y}_{\nu +\mu }(z)$, $\mathring{Y}_{\nu +2\mu }(z)$ of (2.19) having asymptotic behaviour $\mathring{Y}_F(z)=Y_F(z,u^c)$, for $z\rightarrow \infty$, respectively, on sectors $\tau _\nu -\pi<\arg z< \tau _{\nu +1}$, $\tau _\nu<\arg z < \tau _{\nu +\mu +1}$ and $\tau _{\nu +\mu }<\arg z<\tau _{\nu +2\mu +1}$;

• (II,3) $\mathring{L}$, $\mathring{D}$ are the exponents of a fundamental solution $\mathring{Y}(z)=\mathring{G}\left( I+\sum _{j=1}^\infty \mathring{\Psi }_jz^j\right) z^{\mathring{D}}z^{\mathring{L}}$ of (2.19) in Levelt form;

• (II,4) $\mathring{C}_\nu$ connects $\mathring{Y}_\nu (z)=\mathring{Y}(z)\mathring{C}_\nu$.

• (II,5) The Stokes matrices satisfy the vanishing conditions

  $$\begin{aligned}&(\mathbb {S}_\nu )_{ij}=(\mathbb {S}_\nu )_{ji}=0,\quad (\mathbb {S}_{\nu +\mu })_{ij}=(\mathbb {S}_{\nu +\mu })_{ji}=0 \\&\quad \forall ~1\le i\ne j\le n \hbox { such that } u_i^c=u_j^c. \end{aligned}$$

Corollary 2.1

([13]). If $A_{ii}-A_{jj}\not \in \mathbb {Z}\backslash \{0\}$ for every $i\ne j$ such that $u_i^c=u_j^c$, then the formal solution $\mathring{Y}_F(z)$ of (2.19) is unique and coincides with $Y_F(z,u^c)$.

The assumption of Corollary 2.1 will be called partial non-resonance. If it holds, (II,1) says that in order to obtain the essential monodromy data of (1.1) it suffices to compute $\mathring{\mathbb {S}}_\nu$, $\mathring{\mathbb {S}}_{\nu +\mu }$, $\mathring{L}$, $\mathring{C}_\nu$ and $\mathring{D}$ for (2.19), which is simper than (1.1), because $A_{ij}(u^c)=0$ for i, j such that $u_i^c=u_j^c$ . This allows in some cases the explicit computation of monodromy data. An important example with algebro-geometric implications can be found in [14].

Remark 2.1

The following statement, not mentioned in [13], holds.

If (1.1) is an isomonodromic family on the polydisc minus the coalescence locus, in the sense that $dY=\omega Y$ in (2.14)–(2.15) is Frobenius integrable on $\mathbb {D}(u^c)\backslash \Delta$, and if A(u) is holomorphic on $\mathbb {D}(u^c)$, then the vanishing conditions (2.22) hold automatically and (1.1) is isomonodromic on $\mathbb {D}(u^c)$ in the strong sense, namely Theorem 2.2 holds.

I thank the referee for suggesting to write the above statement. The sketch of the proof is as follows: integrability $\mathrm{d}\omega =\omega \wedge \omega$ on $\mathbb {D}(u^c)\backslash \Delta$ implies (2.16), namely

$$\begin{aligned} \frac{\partial A}{\partial u_j}= [\omega _j(u),A],\quad j=1,\ldots ,n;\quad u\in \mathbb {D}(u^c)\backslash \Delta . \end{aligned}$$

(2.23)

We want to prove that $A_{ij}(u)\rightarrow 0$ for $u_i-u_j\rightarrow 0$, for $i\ne j$. From (2.23) and (2.18) we explicitly obtain

$$\begin{aligned} \frac{\partial A_{i\ell }}{\partial u_j} = \frac{ (u_i-u_\ell )A_{ij}A_{j\ell } }{(u_i-u_j)(u_\ell -u_j)}, \quad \hbox { for } j\ne i, \ell \hbox { and } i\ne \ell . \end{aligned}$$

The left-hand side is holomorphic everywhere on $\mathbb {D}(u^c)$ by assumption on A, and so must be the right-hand side. This implies that holomorphically $A_{ij} =O(u_i-u_j)$ for $u_i-u_j\rightarrow 0$. Then, Theorem 2.2 holds and we conclude. $\square$

The difficulty in proving Theorem 2.2 is the analysis of the Stokes phenomenon at $z=\infty$. On the other hand, coalescences does not affect the analysis at the Fuchsian singularity $z=0$, so it is not an issue for the proof of the statements concerning $Y^{(0)}(z,u)$, L , D and $C_\nu$ (as far as the contribution of $Y^{(0)}$ is concerned). See Proposition 17.1 of [13], and the proof of Theorem 4.9 in [25]. For this reason, in the present paper we will not deal with $Y^{(0)}(z,u)$, L , D, $C_\nu$ and (II,3)-(II,4) above.

In Theorem 7.1 we introduce an isomonodromic Laplace transform in order to prove the statements of Theorem2.2above, concerning the Stokes phenomenon, namely (I,1), (I,2), (I,3) and (II,1), (II,2), (II,5).

2.2 Background 2: Laplace transform, connection coefficients and Stokes matrices

In this section, we fix $u\in \mathbb {D}(u^c)\backslash \Delta$. Accordingly, system (1.1) is to be considered as a system not depending on deformation parameters, with leading matrix $\Lambda$ having pairwise distinct eigenvalues, and system (1.4) is equivalent to (1.3), which does not depend on parameters. For simplicity of notations, let us fix for example

$$\begin{aligned} u=u^{0},\quad \hbox { as in Section }2.1.1. \end{aligned}$$

Solutions $Y_\nu (z)$ of (1.1) with canonical asymptotics $Y_F(z)$ ($u=u^{0}$ fixed is not indicated) can be expressed in terms of convergent Laplace-type integrals [5, 31], where the integrands are solutions of the Fuchsian system⁵

$$\begin{aligned} (\Lambda -\lambda )\frac{\mathrm{d} \Psi }{ \mathrm{d}\lambda }= (A+I)\Psi ,\quad I:=\hbox { identity matrix} \end{aligned}$$

(2.24)

Indeed, let $\vec {\Psi }(\lambda )$ be a vector valued function and define

$$\begin{aligned} \vec {Y}(z)=\int _\gamma e^{\lambda z}\vec {\Psi }(\lambda )\mathrm{d}\lambda , \end{aligned}$$

where $\gamma$ is a suitable path. Then, substituting in (1.1), we have

$$\begin{aligned} (z \Lambda +A)\int _\gamma e^{\lambda z}\vec {\Psi }(\lambda )\mathrm{d}\lambda =z\frac{\mathrm{d}}{ \mathrm{d}z}\int _\gamma e^{\lambda z}\vec {\Psi }(\lambda )\mathrm{d}\lambda = z\int _\gamma \lambda e^{\lambda z}\vec {\Psi }(\lambda )\mathrm{d}\lambda . \end{aligned}$$

This implies that

$$\begin{aligned} A \int _\gamma e^{\lambda z}\vec {\Psi }(\lambda )\mathrm{d}\lambda= & {} \int _\gamma \frac{\mathrm{d}(e^{\lambda z}) }{ \mathrm{d}\lambda } ~(\lambda -\Lambda )\vec {\Psi }(\lambda )\mathrm{d}\lambda \nonumber \\= & {} e^{\lambda z}(\lambda -\Lambda )\vec {\Psi }(\lambda )\Bigl |_\gamma -\int _\gamma e^{\lambda z} \left[ (\lambda -\Lambda )\frac{\mathrm{d}\vec {\Psi }(\lambda )}{ \mathrm{d}\lambda }+\vec {\Psi }(\lambda ) \right] \mathrm{d}\lambda . \end{aligned}$$

(2.25)

If $\gamma$ is such that $e^{\lambda z}(\lambda -\Lambda )\vec {\Psi }(\lambda )\Bigl |_\gamma =0$, and if the function $\vec {\Psi }(\lambda )$ solves (2.24), then $\vec {Y}(z)$ solves (1.1).

Multiplying to the left by $(\Lambda -\lambda )^{-1}$, system (2.24) becomes (1.3),

$$\begin{aligned} \frac{\mathrm{d}\Psi }{ \mathrm{d}\lambda }=\sum _{k=1}^n \frac{B_k }{ \lambda -u_k^0}\Psi ,\quad B_k:=-E_k(A+I). \end{aligned}$$

(2.26)

In order to define matrix solutions of of (2.26) as single valued functions, we consider the $\lambda$-plane with branch-cuts. Let $\eta ^{(0)}\in \mathbb {R}$ satisfy

$$\begin{aligned} \eta ^{(0)}\ne \arg (u_j^0-u_k^0) \hbox { mod } \pi ,~~~\hbox { for all }1\le j,k\le n. \end{aligned}$$

(2.27)

We fix parallel cuts $L_k(\eta ^{(0)})$, namely half-lines oriented from $u_k^0$ to $\infty$ in direction $\arg (\lambda -u_k^0)=\eta ^{(0)}$, $1\le k \le n$. See Fig. 2. Conditions (2.27) mean that a cut $L_k$ does not contain another pole $u_j^0$, $j\ne k$. For this reason $\eta ^{(0)}$ is called admissible direction at $u^0$. Then, we choose a branch of the logarithms $\ln (\lambda - u_k^0)=\ln |\lambda - u_k^0|+i\arg (\lambda -u_k^0)$ by

$$\begin{aligned} \eta ^{(0)}-2\pi<\arg (\lambda -u_k^0)<\eta ^{(0)},\quad \quad k=1,\ldots ,n. \end{aligned}$$

(2.28)

Following [4], the $\lambda$-plane with these cuts and choices of the logarithms is denoted by $\mathcal {P}_{\eta ^{(0)}}$. Matrix solutions of (2.26) are well defined as single-valued functions of $\lambda \in \mathcal {P}_{\eta ^{(0)}}$.

Remark 2.2

$\mathcal {P}_{\eta ^{(0)}}$ can be identified with one of the countably many components of

$$\begin{aligned} \mathcal {R}^\prime :=\mathcal {R}( \mathbb {C}\backslash \{u_1^0,\ldots ,u_n^0\})-(\hbox {lift of all half-lines } L_k). \end{aligned}$$

Each component is obtained by a deck transformation starting from one. Fix one component, for example $\mathcal {P}_{\eta ^{(0)}}$, and define 2n letters

$$\begin{aligned}&l_k:=\hbox { cross a lift of } L_k\hbox { from the right}, \\&l_k^{-1}:=\hbox { cross a lift of } L_k\hbox { from the left},\quad k=1,\ldots ,n, \end{aligned}$$

where “right” or “left” refers to the orientation of $L_k$. The other components are reached by crossing the cuts, so that there is a one to one correspondence between finite sequences $\{l_{j_1}^{\pm 1},\ldots ,l_{j_m}^{\pm 1}\}$ not containing successively a $l_k^{\pm 1}$ and $l_k^{\mp 1}$, and components of $\mathcal {R}^\prime$ (here $j_1,\ldots ,j_m\in \{1,\ldots ,n\}$ and $m\in \mathbb {N}$). The relations (2.28) alone do not identify a component of $\mathcal {R}^\prime$ (as incorrectly written in [23], page 387). For example, the word $l_1l_2l_1^{-1}l_2^{-1}$ leads to a new component of $\mathcal {R}^\prime$ where $\arg (\lambda -u_1^0),\ldots ,\arg (\lambda -u_n^0)$ take the same values of the starting component.⁶ I thank the referee for this remark.

Stokes matrices for (1.1), for fixed and pairwise distinct $u_1^0,\ldots ,u_n^0$, can been expressed in terms of connection coefficients of selected solutions of (2.26). The explicit relations have been obtained in [4] for the generic case when all $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n \not \in \mathbb {Z}$; and in [23] for the general case with no restrictions on $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n$ and A.

2.2.1 Selected vector solutions

The Laplace transform involves three types of vector solutions of (2.26), denoted in [23], respectively, by $\vec {\Psi }_k(\lambda )$, $\vec {\Psi }_k^{*}(\lambda )$ and $\vec {\Psi }_k^{(sing)}(\lambda )$ , for $k=1,\ldots ,n$ (in [4] the notation used is $Y_k$ and $Y_k^*$, while $Y_k^{(sing)}$ does not appear, since it reduces to $Y_k$ in the generic case $\lambda ^\prime _k\not \in \mathbb {Z}$). We will not describe here the $\vec {\Psi }_k^{*}(\lambda )$, which play mostly a technical role. Let

$$\begin{aligned}&\mathbb {N}=\{0,1,2,\ldots \} \hbox { integers},\quad \mathbb {Z}_{-}= \{-1,-2,-3,\ldots \}\hbox { negative integers}, \\&\vec {e}_k = \hbox { standard } k\hbox {th unit column vector in } \mathbb {C}^n. \end{aligned}$$

It is proved in [23] that for every $k\in \{1,\ldots ,n\}$ there are at least $n-1$ independent vector solutions holomorphic at $\lambda =u_k^0$. The remaining independent solution is singular at $\lambda =u_k^0$, except for some exceptional cases possibly occurring when $\lambda ^\prime _k\le -2$ is integer. In such cases, there exist n holomorphic solutions at $\lambda =u_k^0$ (such cases never occur if none of the eigenvalues of A is a negative integer). The selected vector solutions $\vec {\Psi }_k$ are obtained as follows.

• If $\lambda ^\prime _k\le -2$ is integer and we are in an exceptional case when there are no singular solutions at $u_k^0$, namely

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda )\equiv 0, \end{aligned}$$

  then $\vec {\Psi }_k$ is the unique analytic solution with the following normalization:

  $$\begin{aligned} \vec {\Psi }_k(\lambda ) =\left( \frac{(-1)^{\lambda ^\prime _k}}{ (-\lambda ^\prime _k-1)!}\vec {e}_k+\sum _{l\ge 1} \vec {b}_l^{~(k)}(\lambda -u_k^0)^l\right) (\lambda -u_k^0)^{-\lambda ^\prime _k-1}. \end{aligned}$$

• In all other cases, there is a solution $\vec {\Psi }_k^{(sing)}$ with singular behaviour at $\lambda =u_k^0$. This is determined up to a multiplicative factor and the addition of an arbitrary linear combination of the remaining $n-1$ regular at $\lambda =u_k^0$ solutions, denoted below with $\hbox {reg}(\lambda -u_k^0)$. In [23], it has the following structure

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda )=\left\{ \begin{array}{cc} \vec {\psi }_k(\lambda )(\lambda -u_k^0)^{-\lambda ^\prime _k-1}+\hbox {reg}(\lambda -u_k^0) ,&{} \lambda ^\prime _k\not \in \mathbb {Z},\\ \\ \vec {\psi }_k(\lambda )\ln (\lambda -u_k^0)+\hbox {reg}(\lambda -u_k^0), &{} \lambda ^\prime _k\in \mathbb {Z}_{-},\\ \\ \dfrac{ P_k(\lambda ) }{ (\lambda -u_k^0)^{\lambda ^\prime _k+1}} +\vec {\psi }_k(\lambda )\ln (\lambda -u_k^0)+\hbox {reg}(\lambda -u_k^0),&{} \lambda ^\prime _k\in \mathbb {N} . \end{array} \right. \end{aligned}$$

  (2.29)

  Here $\vec {\psi }_k(\lambda )$ is analytic at $u_k^0$ and $P_k(\lambda )=\sum _{l=0}^{\lambda ^\prime _k} \vec {b}_l^{~(k)} (\lambda -u_k^0)^l$ is a polynomial of degree $\lambda ^\prime _k$. We choose the following normalization at $\lambda =u_k^0$

  $$\begin{aligned} \left\{ \begin{array}{cc} \vec {\psi }_k(\lambda )=\Gamma (\lambda ^\prime _k+1)\vec {e}_k+\sum _{l\ge 1} \vec {b}_l^{~(k)}(\lambda -u_k^0)^l,&{} \lambda ^\prime _k\not \in \mathbb {Z},\\ \\ \vec {\psi }_k(\lambda )= \left( \dfrac{(-1)^{\lambda ^\prime _k}}{ (-\lambda ^\prime _k-1)!}\vec {e}_k+\sum _{l\ge 1} \vec {b}_l^{~(k)}(\lambda -u_k^0)^l\right) (\lambda -u_k^0)^{-\lambda ^\prime _k-1}&{} \lambda ^\prime _k\in \mathbb {Z}_{-},\\ \\ P_k(\lambda )=\lambda ^\prime _k!~\vec {e}_k +O(\lambda -u_k^0) &{} \lambda ^\prime _k\in \mathbb {N} , \end{array} \right. \end{aligned}$$

  The coefficients $\vec {b}_l^{~(k)}\in \mathbb {C}^n$ are uniquely determined by the normalization. Then the selected vector solutions $\vec {\Psi }_k$ are uniquely defined by⁷

  $$\begin{aligned} \vec {\Psi }_k(\lambda ):=\vec {\psi }_k(\lambda )(\lambda -u_k^0)^{-\lambda ^\prime _k-1} \quad \hbox {for } \lambda ^\prime _k\not \in \mathbb {Z};\quad \quad \vec {\Psi }_k(\lambda ):=\vec {\psi }_k(\lambda ) \quad \hbox {for } \lambda ^\prime _k\in \mathbb {Z}. \end{aligned}$$

  (2.30)

  In case $\lambda ^\prime _k\in \mathbb {N}$, depending on the system, it may exceptionally happen that

  $$\begin{aligned} \vec {\Psi }_k:=\vec {\psi }_k\equiv 0 . \end{aligned}$$

Remark 2.3

Suppose $\lambda _k^\prime \in \mathbb {Z}$. In particular, if $\lambda ^\prime _k\le -2$, suppose we are in the case when $\vec {\Psi }_k^{(sing)}$ is not identically zero. Then

$$\begin{aligned} \vec {\Psi }_k(\lambda ) =\frac{1}{2\pi i} \left( \vec {\Psi }_k^{(sing)}(l_k(\lambda ))-\vec {\Psi }_k^{(sing)}(\lambda )\right) ,\quad \lambda \in \mathcal {P}_{\eta ^{(0)}}, \end{aligned}$$

is the difference of two singular solutions defined on $\mathcal {P}_{\eta ^{(0)}}$. Here, in the notation of Remark 2.2, the function $\vec {\Psi }_k^{(sing)}(l_k(\lambda ))$ is the value at $\lambda \in \mathcal {P}_{\eta ^{(0)}}$ of the analytic continuation of $\vec {\Psi }_k^{(sing)}(\lambda )$ when passing from a prefixed component of $\mathcal {R}^\prime$, in this case $\mathcal {P}_{\eta ^{(0)}}$, to the component associated with the sequence $\{l_k\}$ of only one element. Namely, the analytic continuation for a small loop $(\lambda -u_k^{(0)})\longmapsto (\lambda -u_k^{(0)})e^{2\pi i}$.

2.2.2 Connection coefficients

Above, the behaviour of $\vec {\Psi }_k(\lambda )$ has been described at $\lambda =u_k^0$. The behaviour at any point $\lambda =u_j^0$, for $j=1,\ldots ,n$, will be expressed by linear relations

$$\begin{aligned}&\vec {\Psi }_k(\lambda )=\vec {\Psi }_j^{(sing)}(\lambda ) c_{jk}+\hbox {reg}(\lambda -u_j^0).\nonumber \\&c_{jk}:=0,\quad \forall k=1,\ldots ,n,\quad \hbox { when } \vec {\Psi }_j^{(sing)}(\lambda )\equiv 0 \hbox { (possible only if } \lambda ^\prime _j\in -\mathbb { N}-2).\nonumber \\ \end{aligned}$$

(2.31)

The above relations define the connection coefficients $c_{jk}$. From the definition, we see that $c_{kk}=1$ for $\lambda ^\prime _k\not \in \mathbb { Z}$, while $c_{kk}=0$ for $\lambda ^\prime _k\in \mathbb {Z}$. In case $\lambda ^\prime _k\in \mathbb { N}$, if it happens that $\vec {\Psi }_k\equiv 0$, then $c_{jk}=0$ for any $j=1,..,n$.

Fig. 2

The poles $u_j^0$, $1\le j \le n$, of system (2.26) and plane $\mathcal {P}_{\eta ^{(0)}}$ with branch cuts $L_j$

Proposition 2.4

(see [4] and Propositions 3, 4 of [23]). If A has no integer eigenvalues, then

$$\begin{aligned} \Psi (\lambda )=\Bigr [\vec {\Psi }_1(\lambda )~|~\cdots ~|~\vec {\Psi }_n(\lambda )\Bigr ],~~~~~\lambda \in \mathcal{P}_{\eta ^{(0)}} \end{aligned}$$

(2.32)

(each $\vec {\Psi _k}$ occupies a column) is a fundamental matrix solution of (2.26). Moreover, the matrix $C:=(c_{jk})$ is invertible if and only if A has no integer eigenvalues. If A has integer eigenvalues and $\Psi$ is fundamental, then some $\lambda ^\prime _k\in \mathbb {Z}$.

2.2.3 Laplace transform and stokes matrices in terms of connection coefficients

If $\eta ^{(0)}$ is admissible in the $\lambda$-plane, with respect to the fixed and pairwise distinct $u_1^0,\ldots ,u_n^0$, then

$$\begin{aligned} \arg z=\tau ^{(0)} := 3\pi /2-\eta ^{(0)} \end{aligned}$$

is an admissible direction (2.2) in the z-plane for system (1.1) at the fixed $u=u^0$. We consider the Stokes rays of $\Lambda (u^0))$ as before. For some $\nu \in \mathbb {Z}$, a labelling (2.3) holds, so that

$$\begin{aligned} \tau _\nu<\tau ^{(0)}<\tau _{\nu +1} \quad \Longleftrightarrow \quad \quad \eta _{\nu +1}<\eta ^{(0)}<\eta _\nu ,\quad \quad \eta _\nu :=\frac{3\pi }{2}-\tau _\nu . \end{aligned}$$

(2.33)

In order to keep track of (2.33), we label (2.32) with $\nu$,

$$\begin{aligned} \Psi _\nu (\lambda )=\Bigr [\vec {\Psi }_1(\lambda ~|\nu )~|~\cdots ~|~\vec {\Psi }_n(\lambda ~|\nu )\Bigr ],\quad \lambda \in \mathcal {P}_{\eta ^{(0)}}. \end{aligned}$$

(2.34)

The connection coefficients will be labelled accordingly as $c^{(\nu )}_{jk}$. Also the singular vector solutions will be labelled $\vec {\Psi }_k^{(sing)}(\lambda ~|\nu )$, $\lambda \in \mathcal {P}_{\eta ^{(0)}}$ as above.

The relation between solutions $\vec {\Psi }_k(\lambda ~|\nu )$ or $\vec {\Psi }_k^{(sing)}(\lambda ~|\nu )$ and the columns of $Y_\nu (z)$ is established in [23] for all values of $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n$, and in [4] for non integer values only. It is given by Laplace-type integrals (Proposition 8 of [23])

$$\begin{aligned} \vec {Y}_k(z~|\nu )= & {} \dfrac{1}{2\pi i } \int _{\gamma _k(\eta ^{(0)})} e^{z\lambda } \vec {\Psi }_k^{(sing)}(\lambda ~|\nu ) \mathrm{d}\lambda , \hbox { if } \lambda ^\prime _k\not \in \mathbb {Z}_{-}; \\ \vec {Y}_k(z~|\nu )= & {} \int _{L_k(\eta ^{(0)})} e^{z\lambda } \vec {\Psi }_k(\lambda ~|\nu ) \mathrm{d}\lambda , \hbox { if } \lambda ^\prime _k\in \mathbb {Z}_{-}. \end{aligned}$$

Here, $\gamma _k(\eta ^{(0)})$ is the path coming from $\infty$ along the left side of the oriented $L_k(\eta ^{(0)})$, encircling $u_k^0$ with a small loop excluding all the other poles, and going back to $\infty$ along the right side of $L_k(\eta ^{(0)})$.

The same as (2.34) can be defined for the cut-plane $\mathcal {P}_{\eta ^\prime }$, with an admissible direction $\eta ^\prime$ satisfying

$$\begin{aligned} \eta _{\nu +k\mu ^{(0)}+1}<\eta ^\prime <\eta _{\mu ^{(0)}+k\mu ^{(0)}},\quad k\in \mathbb {Z}, \end{aligned}$$

and will be denoted by $\Psi _{\nu +k\mu ^{(0)}}(\lambda )$, and analogously for the vectors $\vec {\Psi }_k(\lambda ~|\nu +k\mu ^{(0)})$ and $\vec {\Psi }_k^{(sing)}(\lambda ~|\nu +k\mu ^{(0)})$. From the Laplace transforms of $\vec {\Psi }_k(\lambda ~|\nu +k\mu ^{(0)})$ or $\vec {\Psi }_k^{(sing)}(\lambda ~|\nu +k\mu ^{(0)})$, with the paths of integration $\gamma _k(\eta ^\prime )$ or $L_k(\eta ^\prime )$, we receive $Y_{\nu +k\mu ^{(0)}}(z)$.

Introduce in $\{1,2,\ldots ,n\}$ the ordering $\prec$ given by

$$\begin{aligned} j\prec k ~~\Longleftrightarrow ~~ \mathfrak {R}(z(u_j^0-u_k^0))<0 ~\hbox { for } \arg z=\tau ^{(0)},~~~~~i\ne j,~~i,j\in \{1,\ldots ,n\}. \end{aligned}$$

The following important results, proved in theorem 1 of [23] for all values of $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n$, and in the seminal paper [4] in the generic case $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n\not \in \mathbb {Z}$, establish the relation between Stokes matrices and connection coefficients.⁸

Theorem 2.3

Let $u=u^0$ be fixed so that $\Lambda (u^0)$ has pairwise distinct eigenvalues. Let $\eta ^{(0)}$ and $\tau ^{(0)} = 3\pi /2-\eta ^{(0)}$ be admissible for $u^0$ in the $\lambda$-plane and z-plane, respectively. Suppose that the labelling of Stokes rays is (2.3) and (2.33). Then, the Stokes matrices of system (1.1) at $u=u^0$ are given in terms of the connection coefficients $c_{jk}^{(\nu )}$ of system (2.26), according to the following formulae

$$\begin{aligned} \bigl (\mathbb {S}_\nu \bigr )_{jk}= & {} \left\{ \begin{array}{ll} e^{2\pi i \lambda ^\prime _k}\alpha _k ~c^{(\nu )}_{jk}&{} ~~~\hbox { for } j\prec k, \\ \\ 1 &{} ~~~\hbox { for } j =k, \\ \\ 0 &{} ~~~\hbox { for } j\succ k, \end{array} \right. \\ \bigl ( \mathbb {S}_{\nu +\mu ^{(0)}}^{-1}\bigr )_{jk}= & {} \left\{ \begin{array}{ll} 0 &{} ~~~\hbox { for } j\prec k, \\ \\ 1 &{}~~~ \hbox { for } j =k, \\ \\ -e^{2\pi i (\lambda ^\prime _k-\lambda ^\prime _j)}\alpha _k~c^{(\nu )}_{jk} &{} ~~~\hbox { for } j\succ k. \end{array} \right. \end{aligned}$$

where,

$$\begin{aligned} \alpha _k:=(e^{-2\pi i \lambda ^\prime _k}-1)\quad \hbox { if }\lambda ^\prime _k\not \in \mathbb {Z};\quad \quad \alpha _k:=2\pi i \quad \hbox { if }\lambda ^\prime _k\in \mathbb {Z}. \end{aligned}$$

$\square$

In the above discussion, the differential systems do not depend on parameters (u is fixed). The purpose of the present paper is to extend the description of Background 2 to the case depending on deformation parameters and include coalescences in $\mathbb {D}(u^c)$, and then to obtain Theorem 2.2 of Background 1 in terms of an isomonodromic Laplace transform.

3 Equivalence of the isomonodromy deformation equations for (1.1) and (1.4)

The first step in our construction is Proposition 3.1, establishing the equivalence between strong isomonodromy deformations of systems (1.1) and (1.4), for u varying in a $\tau$-cell of $\mathbb {D}(u^c)$. In the specific case of Frobenius manifolds, this fact can be deduced from Chapter 5 of [20]. Here we establish the equivalence in general terms.

According to Theorem 2.1, system (1.1) is strongly isomonodromic in a polydisc $\mathbb {D}(u^0)$ contained in a $\tau$-cell of $\mathbb {D}(u^c)$ if and only if⁹

$$\begin{aligned} dA=\sum _{j=1}^n [\omega _j(u),A]~\mathrm{d}u_j,\quad \quad \omega _j(u)=[F_1(u),E_j], \hbox { given in }(2.18). \end{aligned}$$

(3.1)

On the other hand, system (1.4) is strongly isomonodromic in $\mathbb {D}(u^0)$ by definition ([25], Appendix A), when fundamental matrix solutions in Levelt form at each pole $\lambda =u_j$, $j=1,\ldots ,n$, have constant monodromy exponents and are related to each other by constant connection matrices (not to be confused with the connection coefficients). From [7, 8, 25], the necessary and sufficient condition for the deformation to be strongly isomonodromic (this can also be taken as the definition) is that (1.4) is the $\lambda$-component of a Frobenius integrable Pfaffian system with the following structure

$$\begin{aligned} \mathrm{d}\Psi =P(\lambda ,u) \Psi ,\quad \quad P(\lambda ,u)=\sum _{k=1}^n\frac{B_k(u)}{\lambda -u_k}d(\lambda -u_k) +\sum _{k=1}^n \gamma _k(u) \mathrm{d}u_k. \end{aligned}$$

(3.2)

The integrability condition $dP=P \wedge P$ is the non-normalized Schlesinger system (see Appendix A and [6,7,8, 25, 27, 63])

$$\begin{aligned}&\partial _i\gamma _k-\partial _k\gamma _i=\gamma _i\gamma _k-\gamma _k\gamma _i, \end{aligned}$$

(3.3)

$$\begin{aligned}&\partial _iB_k=\frac{[B_i,B_k]}{u_i-u_k}+[\gamma _i,B_k], \quad i\ne k \end{aligned}$$

(3.4)

$$\begin{aligned}&\partial _iB_i= -\sum _{k\ne i} \frac{[B_i,B_k]}{u_i-u_k}+[\gamma _i,B_i] \end{aligned}$$

(3.5)

Proposition 3.1

The system (3.1) is equivalent to (3.3)–(3.5) if and only if

$$\begin{aligned} \gamma _j(u)\equiv \omega _j(u) \hbox { as in }(2.15) \hbox { and }(2.18),\quad \quad j=1,\ldots ,n. \end{aligned}$$

Namely, (1.1) is strongly isomonodromic in a polydisc on $\mathbb {D}(u^0)$ contained in a $\tau$-cell if and only if (1.4) is strongly isomonodromic.

Proof

See Appendix B. $\square$

4 Schlesinger system on $\mathbb {D}(u^c)$ and vanishing conditions

In this section, Proposition 4.1, we holomorphically extend to $\mathbb {D}(u^c)$ the non-normalized Schlesinger system associated with (1.4), when certain vanishing conditions (4.2) are satisfied. This is the second step to obtain the results of [13] by Laplace transform.

Lemma 4.1

Let A(u) be holomorphic on $\mathbb {D}(u^c)$ and $B_j(u):=-E_j(A(u)+I)$, $j=1,\ldots ,n$.

• i) The vanishing relations

  $$\begin{aligned} {[}B_i(u),B_j(u)]\longrightarrow 0, \quad \hbox { for } u_i-u_j\rightarrow 0\hbox { in } \mathbb {D}(u^c). \end{aligned}$$

  (4.1)

  hold if and only if

  $$\begin{aligned} \bigl (A(u)\bigr )_{ij}\longrightarrow 0, \quad \hbox { for } u_i-u_j\rightarrow 0\hbox { in } \mathbb {D}(u^c). \end{aligned}$$

  (4.2)

• ii) The matrices $\omega _k(u)=[F_1(u),E_k]$ are holomorphic on $\mathbb {D}(u^c)$ if and only if (4.2) holds.

Proof

Let $u^*\in \Delta$, so that for some $i\ne j$ it occurs that $u_i^*=u_j^*$. Since

$$\begin{aligned} B_j=-E_j(A+I)=\begin{pmatrix} 0 &{}&{}&{}0&{}&{}&{}0 \\ \vdots &{}&{}&{}\vdots &{}&{}&{}\vdots \\ -A_{j1}&{}\cdots &{} -A_{j,j-1}&{}-\lambda ^\prime _j-1&{}-A_{j,j+1}&{} \cdots &{}-A_{jn} \\ \vdots &{}&{}&{}\vdots &{}&{}&{}\vdots \\ 0&{}&{}&{}0&{}&{}&{}0 \end{pmatrix}. \end{aligned}$$

(4.3)

it is an elementary computation to check the equivalence between the relation $[B_i(u^*),B_j(u^*)]=0$ and the relation $(A(u^*))_{ij}=0$. Since $[F_1(u),E_k]$ is (2.18), the statement on its analyticity is straightforward. $\square$

Proposition 4.1

Consider a Frobenius integrable Pfaffian system (3.2) on $\mathbb {D}(u^0)$ with

$$\begin{aligned} B_j(u)=-E_j(A(u)+I)\quad \hbox { and }\quad \gamma _j(u)\equiv \omega _j(u)=[F_1(u),E_j] \hbox { in } (2.18). \end{aligned}$$

(4.4)

Assume that A(u) is holomorphic on the whole $\mathbb {D}(u^c)$. Then, system (3.2) is Frobenius integrable on the whole $\mathbb {D}(u^c)$ with holomorphic matrix coefficients if and only if the vanishing conditions (4.2) hold.

Proof

If system (3.2) is integrable on $\mathbb {D}(u^c)$ with holomorphic coefficients $B_k$ and $\gamma _k=\omega _k$, then analyticity of $\omega _k$ with structure (2.18) implies that (4.2) must hold, so that (4.1) holds. Notice that by (4.1), the r.h.sides of (3.4)–(3.5) are holomorphic on $\mathbb {D}(u^c)$. Conversely, suppose that (4.1)–(4.2) hold. By Proposition 3.1, (3.3)–(3.4)–(3.5) are equivalent in $\mathbb {D}(u^0)$ to

$$\begin{aligned} dA=\sum _{j=1}^n [\omega _j(u),A]~\mathrm{d}u_j, \quad u\in \mathbb {D}(u^0). \end{aligned}$$

(4.5)

Now, the l.h.s is well defined and holomorphic on $\mathbb {D}(u^c)$, because so is A(u). The r.h.s. is also analytic on $\mathbb {D}(u^c)$, because of (4.2). Hence, the first part of the Proof of Proposition 3.1 in Appendix B works in the whole $\mathbb {D}(u^c)$, and so the Pfaffian system (3.2) is integrable there. $\square$

For completeness, we also state the following

Proposition 4.2

Let system (3.2) with coefficients (4.4) be integrable on $\mathbb {D}(u^c)\backslash \Delta$ and let the $B_k(u)$ be holomorphic on $\mathbb {D}(u^c)$. Then the vanishing conditions (4.1)–(4.2), hold and the $\omega _k$ are holomorphic on $\mathbb {D}(u^c)$.

Proof

Analogously to the proof of Proposition 3.1, we see that (3.3)–(3.4)–(3.5) on $\mathbb {D}(u^c)\backslash \Delta$ are equivalent to

$$\begin{aligned} dA=\sum _{j=1}^n [\omega _j(u),A]~\mathrm{d}u_j, \quad u\in \mathbb {D}(u^c)\backslash \Delta . \end{aligned}$$

(4.6)

By holomorphy of A(u) on $\mathbb {D}(u^c)$, the r.h.s is well defined, so that also the l.h.s. must be holomorphic on $\mathbb {D}(u^c)$. From (4.6) we proceed as in Remark 2.1, concluding that $A_{ij} =O(u_i-u_j)\rightarrow 0$ holomorphically for $u_i-u_j\rightarrow 0$. The proof can be done also with an argument similar to Remark 6.1. $\square$

5 Selected vector solutions depending on parameters $u\in \mathbb {D}(u^c)$, Theorem 5.1

In this section, we state one main result of the paper, Theorem 5.1, introducing the isomonodromic analogue of the selected and singular vector solutions (2.30) and (2.29). This is the third step required to obtain the results of [13] by Laplace transform.

Preliminarily, we characterize the radius $\epsilon _0>0$ of $\mathbb {D}(u^c)$ in (2.1). The coalescence point $u^c=(u_1^c,\ldots ,u_n^c)$ contains $s<n$ distinct values, say $\lambda _1,\ldots ,\lambda _s$, with algebraic multiplicities $p_1$, ..., $p_s$, respectively ($p_1+\cdots +p_s=n$). Suppose that $\arg z =\tau$ is a direction admissible at $u^c$, as defined in (2.20), and let

$$\begin{aligned} \eta ={3\pi /2}-\tau \end{aligned}$$

be the corresponding admissible direction in the $\lambda$-plane, where we draw parallel half lines $\mathcal {L}_1=\mathcal {L}_1(\eta )$, ..., $\mathcal {L}_s=\mathcal {L}_s(\eta )$ issuing from $\lambda _1$, ..., $\lambda _s$, respectively, with direction $\eta$, as in Fig. 3. Let

$$\begin{aligned} 2\delta _{\alpha \beta }:=\hbox { distance between } \mathcal {L}_\alpha \hbox { and } \mathcal {L}_\beta ,\hbox { for } 1\le \alpha \ne \beta \le s \end{aligned}$$

In formulae, $2\delta _{\alpha \beta }=\min _{\rho >0}|\lambda _\alpha -\lambda _\beta +\rho e^{\sqrt{-1}( 3\pi /2-\tau )}|$. Then, we require that

$$\begin{aligned} \epsilon _0<\min _{1\le \alpha \ne \beta \le n} \delta _{\alpha \beta }. \end{aligned}$$

(5.1)

The bound (5.1) was introduced in [13] in order to prove Theorem 2.2 in Background 1. It implies properties of the Stokes rays as u varies in $\mathbb {D}(u^c)$, described later in Sect. 7. Let

$$\begin{aligned} \mathbb {D}_\alpha := \{\lambda \in \mathbb {C}~|~|\lambda -\lambda _\alpha |\le \epsilon _0\},\quad \alpha =1,\ldots ,s, \end{aligned}$$

be the disc centered a $\lambda _\alpha$ and radius $\epsilon _0$. If $u_j$ is such that $u_j^c=\lambda _\alpha$, the bound (5.1) implies that $u_j$ remains in $\mathbb {D}_\alpha$ as u varies in $\mathbb {D}(u^c)$. Clearly, $\mathbb {D}_\alpha \cap \mathbb {D}_\beta =\emptyset$.

Fig. 3

The figure represents the half lines $\mathcal {L}_\alpha$, $\mathcal {L}_\beta$, etc, for $\alpha ,\beta ,\ldots \in \{1,\ldots ,s\}$, in direction $\eta =3\pi /2-\tau$, the discs centred at the coordinates $\lambda _1,\dots ,\lambda _s$ of the coalescence point $u^c$, and the distances $\delta _{\alpha \beta }$. Also two points $u_i,u_j$ are represented, such that $u_i^c=u_j^c=\lambda _\delta$ for some $\delta \in \{1,\ldots ,s\}$. Important: now $\eta$ refers to $u^c$, differently from Sect. 2.2 and Fig. 2

The Stokes rays of $\Lambda (u^c)$ can be labeled as in (2.21). For a certain $\nu \in \mathbb {Z}$ we have

$$\begin{aligned} \eta _{\nu +1}<\eta<\eta _\nu \quad \Longleftrightarrow \quad \tau _\nu<\tau <\tau _{\nu +1},\quad \quad \eta _\nu =\frac{3\pi }{2}-\tau _\nu . \end{aligned}$$

(5.2)

For each $u\in \mathbb {D}(u^c)$, let $\mathcal {P}_{\eta }=\mathcal {P}_{\eta }(u)$ be the $\lambda$-plane with branch cuts $L_1=L_1(\eta )$, ..., $L_n=L_n(\eta )$ issuing from $u_1,\ldots ,u_n$ and the choice of the logarithms $\ln (\lambda - u_k)=\ln |\lambda - u_k|+i\arg (\lambda -u_k)$, given by

$$\begin{aligned} \eta -2\pi<\arg (\lambda -u_k)<\eta ,\quad \quad k=1,\ldots ,n. \end{aligned}$$

We define the domain (notation $\hat{\times }$ inspired by [33])

$$\begin{aligned} \mathcal {P}_\eta (u)\hat{\times }\mathbb {D}(u^c):=\{(\lambda ,u)~|~ u\in \mathbb {D}(u^c),~\lambda \in \mathcal {P}_\eta (u)\}, \end{aligned}$$

According to Proposition 4.1, for a Pfaffian system (3.2) with coefficients (4.4), defined on a polydisc $\mathbb {D}(u^0)$ contained in a $\tau$-cell of $\mathbb {D}(u^c)$, if A(u) is holomorphic on $\mathbb {D}(u^c)$, then the vanishing conditions (4.2)

$$\begin{aligned} \bigl (A(u)\bigr )_{ij}\longrightarrow 0, \quad \hbox { for } u_i-u_j\rightarrow 0\hbox { in } \mathbb {D}(u^c). \end{aligned}$$

are equivalent to Frobenius integrability on the whole $\mathbb {D}(u^c)$. With this in mind, we state the following

Theorem 5.1

Consider a Pfaffian system,

$$\begin{aligned} \mathrm{d}\Psi =P(\lambda ,u) \Psi ,\quad \quad P(z,u)=\sum _{k=1}^n\frac{B_k(u)}{\lambda -u_k}d(\lambda -u_k) +\sum _{k=1}^n \omega _k(u) \mathrm{d}u_k. \end{aligned}$$

(5.3)

Frobenius integrable on $\mathbb {D}(u^c)$, with matrix coefficients (2.18), and A(u) holomorphic on $\mathbb {D}(u^c)$. Let the radius $\epsilon _0$ be as in (5.1). Then, two classes of vector solutions, holomorphic on $\mathcal {P}_\eta (u)\hat{\times }\mathbb {D}(u^c)$, exist as follows.

The selected solution: $\vec {\Psi }_1(\lambda ,u~|\nu ),~\ldots ~,~\vec {\Psi }_n(\lambda ,u~|\nu )$. Each $\vec {\Psi }_k(\lambda ,u~|\nu )$ is uniquely identified by the local behaviour below for $\lambda \in \mathbb {D}_\alpha$, where $\alpha$ is such that $u_k^c=\lambda _\alpha$. The label $\nu$ keeps track of (5.2).

• $\underline{For \lambda ^\prime _k\in \mathbb {C}\backslash \mathbb {Z} or \lambda ^\prime _k\in \mathbb {Z}_{-}=\{-1,-2,\ldots \}}$,

  $$\begin{aligned} \vec {\Psi }_k(\lambda ,u~|\nu )=\vec {\psi }_k(\lambda ,u~|\nu ) (\lambda -u_k)^{-\lambda ^\prime _k-1},\quad k=1,\ldots ,n, \end{aligned}$$

  (5.4)

  where $\vec {\psi }_k(\lambda ,u~|\nu )$ is holomorphic on $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$ and is represented by a uniformly convergent Taylor expansion with holomorphic on $\mathbb {D}(u^c)$ coefficients:

  $$\begin{aligned} \vec {\psi }_k(\lambda ,u~|\nu )=f_k\vec {e}_k+ \sum _{l=1}^\infty \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l ,\quad \hbox { for } \lambda \rightarrow u_k, \end{aligned}$$

  (5.5)

  The following normalization uniquely identifies $\vec {\Psi }_k$.

  $$\begin{aligned} f_k= \left\{ \begin{array}{ccc} \Gamma (\lambda ^\prime _k+1),&{} \lambda ^\prime _k\in \mathbb {C}\backslash \mathbb {Z}, \\ \dfrac{(-1)^{\lambda ^\prime _k}}{(-\lambda ^\prime _k-1)!}, &{} \lambda ^\prime _k \in \mathbb {Z}_{-}, \end{array} \right. \end{aligned}$$

  (5.6)

• $\underline{For \lambda ^\prime _k\in \mathbb {N}=\{0,1,2,\ldots \}}$,

  $$\begin{aligned} \vec {\Psi }_k(\lambda ,u~|\nu )=\sum _{l=0}^\infty \vec {d}_l^{~(k)}(u)(\lambda -u_k)^l ,\quad \hbox { for } \lambda \rightarrow u_k, \end{aligned}$$

  (5.7)

  is holomorphic on $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$, the Taylor expansion being uniformly convergent with holomorphic coefficients $\vec {d}_l^{~(k)}(u)$. It is uniquely identified by the normalization (5.11) of the singular solution (5.10). Depending on the specific Pfaffian system¹⁰, it may happen that identically

  $$\begin{aligned} \vec {\Psi }_k(\lambda ,u~|\nu )\equiv 0. \end{aligned}$$

The isolated singularities of $\vec {\Psi }_k(\lambda ,u~|\nu )$, if any, are located at $\lambda =u_j$ with $u_j^c=\lambda _\beta$, $\beta \ne \alpha$, and at $\lambda =u_k$ only in case $\lambda ^\prime _k\in \mathbb {C}\backslash \mathbb {Z}$. For $i\ne j$ such that $u_i^c=u_j^c$, $\vec {\Psi }_i(\lambda ,u~|\nu )$ and $\vec {\Psi }_j(\lambda ,u~|\nu )$ are either linearly independent, or at least one of them is identically zero (identity to zero may occur only for $\lambda ^\prime _i$ or $\lambda ^\prime _j$ belonging to $\mathbb {N}$)

The singular solutions: $\vec {\Psi }_1^{(sing)}(\lambda ,u~|\nu ),~\ldots ~,~\vec {\Psi }_n^{(sing)}(\lambda ,u~|\nu )$. Each $\vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu )$ is a solution with an isolated singularity at $\lambda =u_k$, whose singular behaviour is uniquely characterized as follows.¹¹ Let $\mathbb {D}_\alpha$ be identified by $\lambda _\alpha =u_k^c$.

• $\underline{For \lambda ^\prime _k\in \mathbb {C}\backslash \mathbb {Z}}$ [algebraic or logarithmic branch-point],

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu ):= \vec {\Psi }_k(\lambda ,u~|\nu )=\vec {\psi }_k(\lambda ,u~|\nu ) (\lambda -u_k)^{-\lambda ^\prime _k-1}. \end{aligned}$$

• $\underline{For \lambda ^\prime _k\in \mathbb {Z}_{-}}$ [logarithmic branch-point],

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu )&=\vec {\Psi }_k(\lambda ,u~|\nu ) \ln (\lambda -u_k)+\sum _{m\ne k}^* r_m\vec {\Psi }_m(\lambda ,u~|\nu ) \ln (\lambda -u_m)\nonumber \\&\quad + \vec {\phi }_k(\lambda ,u~|\nu ) , \end{aligned}$$

  (5.8)
  $$\begin{aligned}&\underset{\lambda \rightarrow u_k}{=}\vec {\Psi }_k(\lambda ,u~|\nu ) \ln (\lambda -u_k)+\hbox {reg}(\lambda -u_k),\quad \quad r_m\in \mathbb {C}, \end{aligned}$$

  (5.9)

  where $\sum _{m\ne k}^*$ is over all m such that $u_m\in \mathbb {D}_\alpha$ and $\lambda ^\prime _m\in \mathbb {Z}_{-}$. The vector function $\vec {\phi }_k(\lambda ,u~|\nu )$ is holomorphic in $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$.

  In particular, $\underline{\hbox {for } \lambda ^\prime _k\le -2}$, depending on the system, it may happen that there is no solution with singularity in $\mathbb {D}_\alpha$, so that

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu ):=0. \end{aligned}$$

• $\underline{For \lambda ^\prime _k\in \mathbb {N}}$ [logarithmic branch-point and pole],

  $$\begin{aligned} \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu )= \vec {\Psi }_k(\lambda ,u~|\nu )\ln (\lambda -u_k)+\frac{\vec {\psi }_k(\lambda ,u~|\nu )}{(\lambda -u_k)^{\lambda ^\prime _k+1}}, \end{aligned}$$

  (5.10)

  where $\vec {\psi }_k(\lambda ,u~|\nu )$ is holomorphic in $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$,

  $$\begin{aligned} \vec {\psi }_k(\lambda ,u~|\nu )= \Gamma (\lambda ^\prime _k+1)\vec {e}_k+ \sum _{l=1}^\infty \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l ,\quad \hbox { for } \lambda \rightarrow u_i, \end{aligned}$$

  (5.11)

  the Taylor expansion being uniformly convergent and the coefficients $\vec {b}_l^{~(k)}(u)$ holomorphic on $\mathbb {D}(u^c)$.

Let i, j be such that $u_i^c=u_j^c$. Then $\vec {\Psi }_i^{(sing)}(\lambda ,u~|\nu )$ and $\vec {\Psi }_j^{(sing)}(\lambda ,u~|\nu )$ are either linearly independent, or at least one of them is identically zero (identity to zero can be realized only for $\lambda ^\prime _i\le -2$ or $\lambda ^\prime _j\le -2$.)

Proof

See Sect. 6. $\square$

Remark 5.1

Of the coefficients of (5.11), only $b_0^{(i)}(u)$, $b_1^{(i)}(u)$, ..., $b_{\lambda ^\prime _k}^{(k)}(u)$ will be useful later.

Remark 5.2

For $\lambda ^\prime _k\not \in \mathbb {Z}_{-}$, the singular solution $\vec {\Psi }_k^{(sing)}$ is unique, identified by its singular behaviour at $\lambda =u_k$ and the normalization (5.5)–(5.6) when $\lambda ^\prime _k\in \mathbb {C}\backslash \mathbb {Z}$, or by the normalization (5.11) when $\lambda ^\prime _k\in \mathbb {N}$. For $\lambda ^\prime _k\in \mathbb {Z}_{-}$, a singular solution in (5.8) is not unique, but its singular behaviour (5.9) at $\lambda =u_k$ is uniquely fixed by the normalization (5.5)–(5.6). There is a freedom due to the choice of the coefficients $r_m$ and of $\vec {\phi }_k$ in (5.8). See also Remark 6.3.

The singular behaviour of $\vec {\Psi }_k$ at $\lambda =u_j$ is expressed by connection coefficients.

Definition 5.1

The connection coefficients are defined by

$$\begin{aligned} \vec {\Psi }_k(\lambda ,u~|\nu )\underset{\lambda \rightarrow u_j}{=}\vec {\Psi }_j^{(sing)}(\lambda ,u~|\nu ) ~c_{jk}^{(\nu )}~+\hbox {reg}(\lambda -u_j),\quad \quad \lambda \in \mathcal {P}_\eta , \end{aligned}$$

(5.12)

and by

$$\begin{aligned} c_{jk}^{(\nu )}:=0,~\forall k=1,\ldots ,n,\quad \hbox {when } \vec {\Psi }^{(sing)}_j\equiv 0,\hbox { possibly occurring for } \lambda ^\prime _j\in -\mathbb {N}-2. \end{aligned}$$

(5.13)

The uniqueness of the singular behaviour of $\vec {\Psi }_j^{(sing)}$ at $\lambda =u_j$ implies that the $c_{jk}$ are uniquely defined. From the definition, we see that

• If $\lambda ^\prime _k\not \in \mathbb {Z}$, $c^{(\nu )}_{kk}=1$.

• If $\lambda ^\prime _k\in \mathbb {Z}$, $c^{(\nu )}_{kk}=0$.

• If $\lambda ^\prime _k\in \mathbb {N}$ and $\vec {\Psi }_k(\lambda ,u~|\nu )\equiv 0$, then $c^{(\nu )}_{1k}=c^{(\nu )}_{2k}=\dots =c^{(\nu )}_{nk}=0$.

• If $\lambda ^\prime _j\in -\mathbb {N}-2$ and $\vec {\Psi }_j^{(sing)}(\lambda ,u~|\nu )\equiv 0$, then $c^{(\nu )}_{j1}=c^{(\nu )}_{j2}=\dots =c^{(\nu )}_{jn}=0$.

Proposition 5.1

The coefficients in (5.12)–(5.13) are isomonodromic connection coefficients, namely they are independent of $u\in \mathbb {D}(u^c)$. They satisfy the vanishing relations

$$\begin{aligned} c_{jk}^{(\nu )}=0 \quad \hbox { for } j\ne k \hbox { such that } u^c_j=u^c_k. \end{aligned}$$

(5.14)

Proof

See Sect. 6.6. $\square$

6 Proof of Theorem 5.1

Remark on notations: Throughout this section, we work with functions $f=f(\lambda ,u|~\nu )$ defined on $\mathcal {P}_\eta (u)\hat{\times }\mathbb {D}(u^c)$. For simplicity we omit $\nu$ and write $f=f(\lambda ,u)$. Similarly, we write $c_{jk}$ in place of $c_{jk}^{(\nu )}$.

6.1 Fundamental matrix solution of the Pfaffian System

Without loss of generality, we order the eigenvalues so that¹²

$$\begin{aligned}&u_1^c=\dots =u_{p_1}^c=\lambda _1; \quad u^c_{p_1+1}=\dots =u^c_{p_1+p_2}=\lambda _2; \end{aligned}$$

(6.1)

$$\begin{aligned}&u^c_{p_1+p_2+1}=\dots =u^c_{p_1+p_2+p_3}=\lambda _3; \quad \ldots .. \hbox { up to } \quad u^c_{p_1+\cdots +p_{s-1}+1}\nonumber \\&\quad =\dots =u^c_{p_1+\cdots +p_{s-1}+p_s}=\lambda _s. \end{aligned}$$

(6.2)

We analyse first the coalescence of $u_1$, ..., $u_{p_1}$ to $\lambda _1$. Other cases are analogous. We change variables $(u_1,\ldots ,u_n,\lambda )\mapsto (x_1,\ldots ,x_{n+1})$ as follows

$$\begin{aligned} x_{n+1}=\lambda -\lambda _1,\quad \quad x_j=\left\{ \begin{array}{cc} \lambda -u_j,&{} 1\le j \le p_1; \\ u_j-\lambda _1,&{} p_1+1\le j \le n. \end{array} \right. \end{aligned}$$

The inverse transformation is

$$\begin{aligned} \lambda =x_{n+1}+\lambda _1,\quad \quad u_j=\left\{ \begin{array}{c} x_{n+1}-x_j+\lambda _1, \quad 1\le j \le p_1, \\ x_j+\lambda _1, \quad \quad \quad p_1+1\le j \le n. \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} x:=\left( \underbrace{x_1,\ldots ,x_{p_1}}_{p_1}, \underbrace{x_{p_1+1},.\ldots ,x_n}_{n-p_1},~x_{n+1}\right) \equiv \left( \underbrace{x_1,\ldots ,x_{p_1}}_{p_1}, \varvec{x}^\prime ,x_{n+1}\right) , \end{aligned}$$

where $\varvec{x}^\prime :=(x_{p_1+1},.\ldots ,x_n)$. We are interested in the behaviour of solutions for

$$\begin{aligned} x \longrightarrow (\underbrace{0,~0,\ldots ,0}_{p_1},~\varvec{x}^\prime ,0), \end{aligned}$$

corresponding to

$$\begin{aligned} u_1\rightarrow \lambda _1,~\dots ,~u_{p_1}\rightarrow \lambda _1,\quad \hbox { and } \lambda \rightarrow \lambda _1 \end{aligned}$$

namely $u_i-u_j\rightarrow 0$ , $i\ne j$ and $\lambda -u_i\rightarrow 0$, for $i, j\in \{1,\ldots ,p_1\}$. The Pfaffian system (5.3) in variables x, with Fuchsian singularities at $x_1=0, \dots ,x_{p_1}=0$, becomes

$$\begin{aligned} \mathrm{d} \Psi =P(x)\Psi ,\quad \quad P(x)=\sum _{j=1}^{p_1} \frac{P_j(x)}{x_j} \mathrm{d}x_j + \sum _{j=p_1+1}^{n+1} \widehat{P}_j(x)\mathrm{d}x_j \end{aligned}$$

(6.3)

where

$$\begin{aligned}&\frac{P_j(x)}{x_j}=\frac{B_j(x)}{x_j}-\omega _j(x),\quad 1\le j \le p_1, \\&\widehat{P}_j(x)= \frac{B_j(x)}{x_j-x_{n+1}}+\omega _j(x),\quad p_1+1 \le j \le n,\\&\widehat{P}_{n+1}(x)=\sum _{j=p_1+1}^n\frac{B_j(x)}{x_{n+1}-x_j}+\sum _{j=1}^{p_1}\omega _j(x) \end{aligned}$$

The Pfaffian system is assumed integrable with holomorphic in $\mathbb {D}(u^c)$ coefficients, therefore $P_1(x),\ldots ,P_{p_1}(x)$ and $\widehat{P}_{p_1+1}(x),\ldots ,\widehat{P}_{n+1}(x)$ are holomorphic at $(\underbrace{0,\dots ,0}_{p_1},~\varvec{x}^\prime ,0)$, for $\varvec{x}^\prime$ varying as $u_{p_1+1},\dots ,u_n$ vary in $\mathbb {D}(u^c)$.

Remark 6.1

The commutation relations (4.1) at $u=(\underbrace{\lambda _1,\dots ,\lambda _1}_{p_1},\varvec{u}^\prime )$, where $\varvec{u}^\prime :=(u_{p_1+1},\dots ,u_n)$, are

$$\begin{aligned}{}[B_i(\lambda _1,\dots ,\lambda _1,\varvec{u}^\prime ),B_j(\lambda _1,\dots ,\lambda _1,\varvec{u}^\prime )]=0,\quad 1\le i \ne j \le p_1. \end{aligned}$$

(6.4)

They also follow from the integrability condition $dP(x)=P(x)\wedge P(x)$ of (6.3), which implies

$$\begin{aligned} \frac{\partial }{\partial x_i} \left( \frac{P_j}{x_j}\right) -\frac{\partial }{\partial x_j} \left( \frac{P_i}{x_i}\right) -\frac{P_iP_j-P_j P_i}{x_ix_j}=0,\quad \quad 1\le i\ne j \le p_1. \end{aligned}$$

Let $\hat{\varvec{k}}=(k_1,\ldots ,k_{p_1})$, and write $\hat{\varvec{l}}\le \hat{\varvec{k}}$ if $k_i\le l_i$ for all $i\in \{1,\ldots ,p_1\}$. The Taylor convergent series $P_i(x)= \sum _{k_1+\cdots +k_{p_1}\ge 0} P_{i,\hat{\varvec{k}}}(\varvec{x}^\prime ,x_{n+1}) x_1^{k_1}\cdots x_{p_1}^{k_{p_1}},$ has coefficients $P_{i,\widehat{\varvec{k}}}(\varvec{x}^\prime ,x_{n+1})$ holomorphic of $\varvec{x}^\prime$, $x_{n+1}$. The integrability condition becomes [63]

$$\begin{aligned} k_jP_{i,\hat{\varvec{k}}}-k_iP_{j,\hat{\varvec{k}}}+\sum _{\varvec{0}\le \hat{\varvec{l}}\le \hat{\varvec{k}}}[P_{i,\hat{\varvec{l}}},P_{j,\hat{\varvec{k}}-\hat{\varvec{l}}}]=0,\quad 1\le i\ne j \le p_1. \end{aligned}$$

(6.5)

In particular, $P_{i,\hat{\varvec{0}}}(\varvec{x}^\prime ,x_{n+1})=B_i(\underbrace{\lambda _1,\dots ,\lambda _1}_{p_1},\varvec{u}^\prime )$ for $\hat{\varvec{k}}=\hat{\varvec{0}}$, so that (6.5) reduces to (6.4). $\square$

Let us define Jordan matrices

$$\begin{aligned} \widehat{T}^{(j)}&=\hbox {diag}(0,\dots ,0,\underbrace{{-1-\lambda ^\prime _j}}_{{\mathrm{position }} j},0,\dots ,0), \quad \quad \hbox { for } \lambda ^\prime _j\ne -1. \end{aligned}$$

(6.6)

$$\begin{aligned} \widehat{T}^{(j)}&:=J^{(j)}:=\begin{pmatrix} 0 &{}&{}0~\cdots &{}&{}0\\ \vdots &{}\ddots &{}&{} &{} \vdots \\ 0 &{} ~\cdots ~&{} 0~\cdots &{}r_{m_j}^{(j)}&{}0\\ \vdots &{}&{}&{}\ddots &{} \vdots \\ 0 &{}&{}0~\cdots &{}&{}0 \end{pmatrix}\quad \longleftarrow \hbox { row } j, \quad \hbox {for } \lambda ^\prime _j=-1,\nonumber \\ r_{m_j}^{(j)}&:=1,\quad \hbox { is the only non-zero entry in position } (j,m_j),\hbox { with } m_j\ge p_1+1. \end{aligned}$$

(6.7)

Lemma 6.1

Under the assumptions of Theorem 5.1, for every $j\in \{1,\ldots ,n\}$ there exists a holomorphically invertible matrix $G^{(j)}(u)$ on $\mathbb {D}(u^c)$ reducing $B_j(u)$ to constant Jordan form. Moreover, $B_1(u^c),\ldots ,B_{p_1}(u^c)$ are simultaneously reducible to $\widehat{T}^{(1)}, \ldots , \widehat{T}^{(p_1)}$, respectively.

Proof

For every $j\in \{1,\ldots ,n\}$, the Schlesinger system (3.3)–(3.5) implies the Frobenius integrability (on $\mathbb {D}(u^c)$) of the the linear Pfaffian system (see Corollary 9.1, Appendix A)

$$\begin{aligned} \frac{\partial G^{(j)}}{\partial u_k} =\left( \frac{B_k}{u_k-u_j}+\gamma _k\right) G^{(j)}, \quad k\ne j,\quad \quad \frac{\partial G^{(j)}}{\partial u_j}=-\sum _{k\ne j} \left( \frac{B_k}{u_k-u_j}+\gamma _k\right) G^{(j)} \end{aligned}$$

(6.8)

From (3.4)–(3.5) and the above, we receive $\partial _k \bigl ((G^{(j)})^{-1} B_j G^{(j)}\bigr )=0$, $k=1,\ldots ,n$, for a holomorphic on $\mathbb {D}(u^c)$ fundamental matrix solution $G^{(j)}(u)$. Thus, up to $G^{(j)}\mapsto G^{(j)}\mathcal {G}^{(j)}$, $\mathcal {G}^{(j)}\in GL(n,\mathbb {C})$, we can choose $G^{(j)}(u)$ which puts $B_j$ in constant Jordan form. If we consider each $B_j$ separately, now for $j\in \{1,\ldots ,p_1\}$, it is straightforward that the Jordan forms are the matrices $\widehat{T}^{(j)}$.¹³ An elementary computation shows that $B_1(u^c),\ldots ,B_{p_1}(u^c)$ are actually reducible to $\widehat{T}^{(1)},.\ldots ,\widehat{T}^{(1)}$ simultaneously,¹⁴ because only the jth row of $B_j(u^c)$ is non-zero, and by (4.1) the first $p_1$ entries of this row are zero, except for the (j, j)-entry equal to $-\lambda ^\prime _j-1$. Namely,

$$\begin{aligned} B_j(u^c)=\begin{pmatrix} 0 ~0&{}&{}&{}&{} \cdots &{} 0 \\ \vdots &{}&{}&{}&{} &{} \vdots \\ \mathbf{0 ~0}&{}-\lambda ^\prime _j-1 &{} \mathbf{0 ~0}&{}~ -A_{j,p_1+1}^{(j)}(u^c)&{} \cdots &{} -A_{j,n}(u^c) \\ \vdots &{}&{}&{}&{}&{} \\ 0~0 &{}&{}&{}&{} \cdots &{} 0 \end{pmatrix}\quad \longleftarrow \hbox { row } j. \end{aligned}$$

$\square$

Remark 6.2

As in Lemma 6.1, $B_1(u^c),\ldots ,B_{p_1}(u^c)$ are reducible simultaneously to their respective Jordan forms, $B_{p_1+1}(u^c),\ldots ,B_{p_1+p_2}(u^c)$ are reducible simultaneously to their respective Jordan forms, and so on up to $B_{p_1+\cdots +p_{s-1}+1}(u^c),\ldots ,B_{p_1+}{\cdots +p_s}(u^c)$.

For short, let $\varvec{p}_1:=(1,\ldots ,p_1)$. Without loss of generality, we label $u_1,\ldots ,u_{p_1}$ so that

$$\begin{aligned} \lambda ^\prime _j\in \mathbb {C}\backslash \mathbb {Z}, \quad \hbox { for } 1\le j \le q_1, \quad \quad \lambda ^\prime _j\in \mathbb {Z}, \quad \hbox { for } q_1+1\le j \le p_1. \end{aligned}$$

If all $\lambda ^\prime _j\in \mathbb {Z}$, then $q_1=0$, if all $\lambda ^\prime _j\not \in \mathbb {Z}$, then $q_1=p_1$. The first and fundamental step to achieve Theorem 5.1 is the following

Theorem 6.1

In the assumptions of Theorem 5.1, the Pfaffian system (5.3) admits the fundamental matrix solution

$$\begin{aligned}&\Psi ^{(\varvec{p}_1)}(\lambda ,u)=G^{(\varvec{p}_1)} U^{(\varvec{p}_1)}(\lambda ,u)\cdot \prod _{l=1}^{p_1} (\lambda -u_l)^{\widehat{T}^{(l)}}\cdot \prod _{j=q_1+1}^{p_1}(\lambda -u_j)^{\widehat{R}^{(j)}},\nonumber \\&\quad (\lambda ,u)\in \mathcal {P}(u)\hat{\times }\mathbb {D}(u^c), \end{aligned}$$

(6.9)

where $G^{(\varvec{p}_1)}$ is a constant invertible matrix simultaneously reducing $B_1(u^c), \ldots , B_{p_1}(u^c)$ to $\widehat{T}^{(1)}, \ldots , \widehat{T}^{(p_1)}$ as in (6.6)–(6.7). The matrix function $U^{(\varvec{p}_1)}(\lambda ,u)$ is holomorphic in $\mathbb {D}_1\times \mathbb {D}(u^c)$ with convergent expansion

$$\begin{aligned}&U^{(\varvec{p}_1)}(\lambda ,u)= I+ \\&\quad +\sum _{\varvec{k}>0,~k_{1}+\cdots +k_{p_1}\ge 0}\Bigl [U^{(\varvec{p}_1)}_{\varvec{k}} \cdot (u_{p_1+1}-u_{p_1+1}^c)^{k_{p_1+1}}\cdots (u_n-u_n^c)^{k_n} (\lambda -\lambda _1)^{k_{n+1}}\Bigr ] \\&\qquad (\lambda -u_1)^{k_1}~\cdots ~(\lambda -u_{p_1})^{k_{p_1}}, \end{aligned}$$

and constant matrix coefficient $U^{(\varvec{p}_1)}_{\varvec{k}}$. Here $\varvec{k}:=(k_1,\ldots ,k_n,k_{n+1})$, $k_j\ge 0$, and $\varvec{k}>0$ means that at least one $k_j>0$ ($j=1,\ldots ,n+1$). The exponents $\widehat{R}^{(q_1+1)},\dots , \widehat{R}^{(p_1)}$ are constant nilpotent matrices.

• If $\lambda ^\prime _j=-1$,

  $$\begin{aligned} \widehat{R}^{(j)}=0. \end{aligned}$$

  (6.10)

• If $\lambda ^\prime _j\in \mathbb {N}=\{0,1,2,\ldots \}$, only the entries $\widehat{R}^{(j)}_{mj}=:r_m^{(j)}$, for $m=1,\ldots ,n$ and $m \ne j$, are possibly non zero, namely

  $$\begin{aligned} \widehat{R}^{(j)}=\left[ \vec {0}~\left| ~\cdots ~\left| ~\vec {0}~\left| ~ \sum _{m\ne j, m=1}^n r^{(j)}_m \vec {e}_m ~\right| ~\vec {0}~\right| ~\cdots ~\right| ~\vec {0} \right] , \end{aligned}$$

  (6.11)

  where only the jth column is possibly non-zero.

• If $\lambda ^\prime _j\in -\mathbb {N}-2=\{-2,-3,\ldots \}$, only the entries $\widehat{R}^{(j)}_{jm}=:r_m^{(j)}$, for $m=1,\ldots ,n$ and $m\ne j$, are possibly non zero, namely

  $$\begin{aligned} \widehat{R}^{(j)}=\begin{pmatrix} 0 &{}\cdots &{}&{}&{}&{} \cdots &{} 0 \\ \vdots &{}&{}&{}&{}&{} &{} \vdots \\ r_1^{(j)} &{} \cdots &{} r_{j-1}^{(j)} &{} 0 &{} r_{j+1}^{(j)}&{} \cdots &{} r_n^{(j)} \\ \vdots &{}&{}&{}&{}&{} &{} \vdots \\ 0 &{} \cdots &{}&{}&{}&{} \cdots &{} 0 \end{pmatrix}\quad \longleftarrow \hbox { row } j\hbox { is possibly non zero }. \end{aligned}$$

  (6.12)

The exponents $\widehat{T}^{(l)}$ and $R^{(j)}$ satisfy the following commutation relations

$$\begin{aligned}&[\widehat{T}^{(i)},\widehat{T}^{(j)}]=0, \quad i,j=1,\ldots ,p_1; \end{aligned}$$

(6.13)

$$\begin{aligned}&[\widehat{R}^{(j)},\widehat{R}^{(k)}]=0, \quad [\widehat{T}^{(i)},\widehat{R}^{(j)}]=0, \quad i=1,\ldots ,p_1,\quad i\ne j,\quad j,k=q_1+1,\ldots ,p_1. \end{aligned}$$

(6.14)

By analytic continuation, $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$ defines an analytic function on the universal covering of $\mathcal {P}_\eta (u)\hat{\times } \mathbb {D}(u^c)$. Another representation of (6.9) will be given in (6.24).

Proof

We apply the results of [63] at the point $x= x^c:=(\underbrace{0,~0,\ldots ,0}_{p_1},~\varvec{x}_c^\prime ,0)$, with $\varvec{x}_c^\prime :=(x^c_{p_1+1},.\ldots ,x_n^c)$, corresponding to $u=u^c$ and $\lambda =\lambda _1$, where $x^c_j= u_j^c-\lambda _1$, $j=p_1+1, \ldots , n$. By Theorem 7 of [63], the Pfaffian system (6.3) admits a fundamental matrix solution

$$\begin{aligned} \Psi ^{(\varvec{p}_1)}(\lambda ,u)=U_0~ U(x)~ Z(x),\quad \quad Z(x)= \prod _{j=1}^{p_1} x_l^{A_j} \prod _{j=1}^{p_1} x_l^{Q_j} ,\quad \quad \det U_0\ne 0, \end{aligned}$$

(6.15)

for certain matrices $A_j$ which are simultaneous triangular forms of $B_1(u^c),\ldots , B_{p_1}(u^c)$. While in [63] a lower triangular form is considered, we equivalently use the upper triangular one. The matrices $Q_j$ will be described below. The matrix $U(x)=V(x)\cdot W(x)$ has structure

$$\begin{aligned}&V(x)= I+\sum _{\varvec{k}>0,~k_{p_1+1}+\cdots +k_{n+1}>0}V_{\varvec{k}}~ x_1^{k_1}~\cdots ~x_{p_1}^{k_{p_1}}~(x_{p_1+1}-x^c_{p_1+1})^{k_{p_1+1}}\\&\quad \cdots (x_n-x^c_n)^{k_n}\cdot x_{n+1}^{k_{n+1}} \\&W(x)=I+\sum _{k_{1}+\cdots +k_{p_1}>0}W_{k_1,\ldots ,k_{p_1}}~ x_1^{k_1}~\cdots ~x_{p_1}^{k_{p_1}}. \end{aligned}$$

The constant matrix coefficients $V_{\varvec{k}}$, $W_{k_1,\ldots ,k_{p_1}}$ can be determined [63] from the constant matrix coefficients $P_{i,\varvec{k}}$ in the Taylor expansion¹⁵ of the $P_j(x)$ and $\widehat{P}_j(x)$. Recall that $x_j= \lambda -u_j$, $1\le j\le p_1$, and $x_{n+1}=\lambda -\lambda _1$. Moreover, for $p_1+1\le j \le n$, we have $x_j-x_j^c= (u_j-\lambda _1)-(u_j^c-\lambda _1)=u_j-u_j^c$. Thus, restoring variables $(\lambda ,u)$, we have

$$\begin{aligned}&V(\lambda ,u)=I+ \\&\quad +\sum _{ k_{p_1+1}+\cdots +k_{n+1}>0 }\Bigl [V_{\varvec{k}} (u_{p_1+1}-u_{p_1+1}^c)^{k_{p_1+1}}\cdot \ldots \cdot (u_n-u_n^c)^{k_n}\cdot (\lambda -\lambda _1)^{k_{n+1}}\Bigr ]\\&\qquad (\lambda -u_1)^{k_1}~\cdots ~(\lambda -u_{p_1})^{k_{p_1}}, \\&W(\lambda , u_1,\ldots ,u_{p_1})=I+\sum _{k_{1}+\cdots +k_{p_1}>0}W_{k_1,\ldots ,k_{p_1}}~ (\lambda -u_1)^{k_1}~\cdot \ldots \cdot ~(\lambda -u_{p_1})^{k_{p_1}}. \end{aligned}$$

Therefore, the matrices appearing in the statement are $G^{(\varvec{p}_1)}:=U_0$ and $U^{(\varvec{p}_1)}(\lambda ,u):= V(\lambda ,u)W(\lambda ,u)$, which is holomorphic for $(\lambda , u)\in \mathbb {D}_1\times \mathbb {D}(u^c)$.

We show that the exponents $A_j$ and $Q_j$ are, respectively, $\widehat{T}^{(j)}$ in (6.6)–(6.7) and $\widehat{R}^{(j)}$ in (6.10)–(6.11)–(6.12). According to [63] (see theorems 2 and 5), the matrix function $G^{(\varvec{p}_1)}\cdot U^{(\varvec{p}_1)}(\lambda ,u)$ in (6.9) provides the gauge transformation

$$\begin{aligned} \Psi =G^{(\varvec{p}_1)}\cdot U^{(\varvec{p}_1)}(\lambda ,u)Z\underset{\hbox {in notation of [63]}}{\equiv }U_0U(x) Z, \end{aligned}$$

which brings (6.3) to the reduced form (being “reduced” is defined in [63])

$$\begin{aligned} dZ=\sum _{j=1}^{p_1}\frac{Q_j(x)}{x_j}~Z,\quad \quad Q_j(x)=A_j +\sum _{\widehat{\varvec{k}}>0} Q_{\widehat{\varvec{k}},j} x_1^{k_1}\cdots x_{p_1}^{k_{p_1}}, \end{aligned}$$

where the notation $\widehat{\varvec{k}}=(k_1,\ldots ,k_{p_1})>0$ means at least one $k_l>0$. From [63], we have the following.

• The $A_j$ are simultaneous triangular forms of $B_1(u^c),\ldots ,B_{p_1}(u^c)$. Thus, by Lemma 6.1, they can be taken to be

  $$\begin{aligned} A_j =\widehat{T}^{(j)} \hbox { as in } (6.6)-(6.7), j=1,\ldots ,p_1. \end{aligned}$$

• The $Q_{\widehat{\varvec{k}},j}$ satisfy diag$(Q_{\widehat{\varvec{k}},j})=0$, while the entry $(\alpha ,\beta )$ for $\alpha \ne \beta$ satisfies

  $$\begin{aligned} (Q_{\widehat{\varvec{k}},j})_{\alpha \beta } \ne 0 \quad \hbox { only if } \quad (\widehat{T}^{(j)})_{\alpha \alpha }-(\widehat{T}^{(j)})_{\beta \beta }= k_j\ge 0,\quad \hbox { for all } j=1,\ldots ,p_1. \end{aligned}$$

Taking into account the particular structure (6.6)–(6.7), the above condition can be satisfied only for

$$\begin{aligned} \widehat{\varvec{k}}=\left( \underbrace{0,\ldots ,0}_{q_1},\underbrace{0,\ldots ,0,k_j,0,\ldots ,0}_{p_1-q_1}\right) ,\quad k_j=|\lambda ^\prime _j+1|\ge 1 \hbox { in position } j, \end{aligned}$$

because

$$\begin{aligned} (\widehat{T}^{(j)})_{\alpha \alpha }-(\widehat{T}^{(j)})_{\beta \beta }&=-\lambda ^\prime _j-1\ge 1 \quad \hbox { when } \lambda ^\prime _j\in -\mathbb {N}-2\quad \hbox { and } \alpha =j \quad (\beta \ne j), \end{aligned}$$

(6.16)

$$\begin{aligned} (\widehat{T}^{(j)})_{\alpha \alpha }-(\widehat{T}^{(j)})_{\beta \beta }&=\lambda ^\prime _j+1\ge 1 \quad \hbox { when } \lambda ^\prime _j\in \mathbb {N}\quad \hbox { and } \beta =j \quad (\alpha \ne j). \end{aligned}$$

(6.17)

This can occur only for $j=q_1+1,\ldots ,p_1$. Thus

$$\begin{aligned} Q_{\widehat{\varvec{k}},j}=0,\quad j=1,\ldots ,q_1,\quad \quad Q_{\widehat{\varvec{k}},j}= \widehat{R}^{(j)} \hbox { in } (6.10)-(6.11)-(6.12),\quad j=q_1+1,\ldots ,p_1. \end{aligned}$$

(6.18)

In conclusion, the reduced form turns out to be

$$\begin{aligned} dZ=\left[ \sum _{j=1}^{p_1}\left( \frac{\widehat{T}^{(j)}+\widehat{R}^{(j)} x^{k_j}}{x_j}\right) \right] Z,\quad \quad \widehat{R}^{(1)}=\dots =\widehat{R}^{(q_1)}=0. \end{aligned}$$

(6.19)

Its integrability implies the commutation relations. Indeed, the compatibility $\partial _i\partial _j Z=\partial _j\partial _i Z$, $i\ne j$, holds if and only if

$$\begin{aligned}&\frac{[\widehat{T}^{(j)},\widehat{T}^{(i)}]}{x_ix_j}+[\widehat{R}^{(j)},\widehat{R}^{(i)}]x_i^{k_i-1}x_j^{k_j-1} +[\widehat{T}^{(j)},\widehat{R}^{(i)}]x_i^{k_i-2}\\&\quad + [\widehat{R}^{(j)},\widehat{T}^{(i)}]x_j^{k_j-2}=0, \quad \quad 1\le i\ne j \le p_1. \end{aligned}$$

Keeping into account that $\widehat{R}^{(1)}=\dots =\widehat{R}^{(q_1)}=0$, the above holds if and only if (6.13)–(6.14) hold.

The last to be checked is that a fundamental matrix of (6.19) is Z(x) in (6.15), namely

$$\begin{aligned} Z(x)= \prod _{l=1}^{p_1} x_l^{\widehat{T}^{(l)}} \prod _{j=q_1+1}^{p_1} x_l^{\widehat{R}^{(j)}}. \end{aligned}$$

It suffices to verify this by differentiating Z(x), keeping into account the commutation relations (6.13)–(6.14) and the formula $\partial _i x_i^M=(M/x_i)x_i^M$, for a constant matrix M. For $i=1,\ldots ,q_1$, we receive

$$\begin{aligned} \frac{\partial }{\partial x_i} Z(x) = \frac{\widehat{T}^{(i)}}{x_i} Z(x). \end{aligned}$$

For $i=q_1+1,\ldots ,p_1$, we receive

$$\begin{aligned} \frac{\partial }{\partial x_i} Z(x)&= \frac{T^{(i)}}{x_i} Z(x)+ \Bigl (\prod _{l=1}^{p_1} x_l^{\widehat{T}^{(l)}} \Bigr )\frac{\widehat{R}^{(i)}}{x_i} \Bigl (\prod _{j=q_1+1}^{p_1} x_l^{\widehat{R}^{(j)}}\Bigr ) \\&=\frac{\widehat{T}^{(i)}}{x_i} Z(x)+ \Bigl (\prod _{l=1}^{i-1} x_l^{\widehat{T}^{(l)}} \Bigr ) \frac{x_i^{\widehat{T}^{(i)}}\widehat{R}^{(i)}}{x_i} \Bigl ( \prod _{l=i+1}^{p_1} x_l^{\widehat{T}^{(l)}} \Bigr ) \Bigl ( \prod _{j=q_1+1}^{p_1} x_l^{\widehat{R}^{(j)}} \Bigr )=(**). \end{aligned}$$

Now, recalling that $k_i=|\lambda ^\prime _i+1|$ and (6.16)–(6.17), we see that $x_i^{\widehat{T}^{(i)}}\widehat{R}^{(i)}x_i^{-\widehat{T}^{(i)}}= \widehat{R}^{(i)}x_i^{k_i}$. Therefore,

$$\begin{aligned} (**)=\frac{\widehat{T}^{(i)}}{x_i} Z(x)+ \frac{\widehat{R}^{(i)} x_i^{k_i}}{x_i} \Bigl ( \prod _{l=1}^{p_1} x_l^{\widehat{T}^{(l)}} \Bigr ) \Bigl ( \prod _{j=q_1+1}^{p_1} x_l^{\widehat{R}^{(j)}} \Bigr )= \frac{\widehat{T}^{(i)}+\widehat{R}^{(i)} x_i^{k_i}}{x_i} Z(x), \end{aligned}$$

as we wanted to prove.

Finally, the fact that $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$ has analytic continuation on the universal covering of $\mathcal {P}_\eta (u)\hat{\times } \mathbb {D}(u^c)$ follows from general results in the theory of linear Pfaffian systems [28, 32, 63]. $\square$

It is convenient to introduce a slight change of the exponents. Without loss in generality, we can label $u_1,\ldots ,u_{p_1}$ in such a way that, for some $q_1,c_1\ge 0$ integers, the following ordering of eigenvalues of A holds:

$$\begin{aligned} \underline{\lambda ^\prime _1,~\dots ,~\lambda ^\prime _{q_1}\in \mathbb {C}\backslash \mathbb {Z},\quad \quad \lambda ^\prime _{q_1+1},~\dots ,~\lambda ^\prime _{q_1+c_1}\in \mathbb {Z}_{-}, \quad \quad \lambda ^\prime _{q_1+c_1+1},~\dots ,~\lambda ^\prime _{p_1}\in \mathbb {N}.} \end{aligned}$$

Clearly, $0\le q_1\le p_1$, $0\le c_1\le p_1$ and $0\le q_1+c_1\le p_1$. We define new exponents.

• For $\lambda ^\prime _j\ne -1$,

  $$\begin{aligned} T^{(j)}:=\widehat{T}^{(j)},\quad j=1,\ldots ,p_1;\quad \quad \quad R^{(j)}:=\widehat{R}^{(j)},\quad j=q_1+1,\ldots ,p_1. \end{aligned}$$

  (6.20)

• For $\lambda ^\prime _j= -1$ (so $j\in \{q_1+1,\ldots ,q_1+c_1\}$),

  $$\begin{aligned} T^{(j)}:=0 ,\quad R^{(j)}:=\underbrace{J^{(j)}}_{{\mathrm{in}} (6.7)}=\begin{pmatrix} 0 &{}&{}0~\cdots &{}&{}0 \\ \vdots &{}\ddots &{}&{} &{} \vdots \\ 0 &{} ~\cdots ~&{} 0~\cdots &{}r_{m_j}^{(j)}&{}0 \\ \vdots &{}&{}&{}\ddots &{} \vdots \\ 0 &{}&{}0~\cdots &{}&{}0 \end{pmatrix} \longleftarrow \hbox { row } j, \quad r_{m_j}^{(j)}=1. \end{aligned}$$

  (6.21)

  Recall that $m_j\ge p_1+1$.

This new definitions allow to treat together the case $\lambda ^\prime _j\in -\mathbb {N}-2$ and the case $\lambda ^\prime _j=-1$.

Lemma 6.2

With the definition (6.20)–(6.21), the following relations hold.

$$\begin{aligned}&[T^{(i)},T^{(j)}]=0, \quad i,j=1,\ldots ,p_1; \end{aligned}$$

(6.22)

$$\begin{aligned}&[R^{(j)},R^{(k)}]=0, \quad [T^{(i)},R^{(j)}]=0, \quad i=1,\ldots ,p_1,\quad i\ne j,\quad j,k=q_1+1,\ldots ,p_1, \end{aligned}$$

(6.23)

Proof

The equivalence between (6.13)–(6.14) and (6.22)–(6.23) is straightforward. $\square$

Corollary 6.1

In Theorem 6.1, the fundamental matrix solution (6.9) is

$$\begin{aligned} \Psi ^{(\varvec{p}_1)}(\lambda ,u)=G^{(\varvec{p}_1)}\cdot U^{(\varvec{p}_1)}(\lambda ,u)\cdot \prod _{l=1}^{p_1} (\lambda -u_l)^{T^{(l)}}\cdot \prod _{j=q_1+1}^{p_1}(\lambda -u_j)^{R^{(j)}}, \end{aligned}$$

(6.24)

where the exponents are defined in (6.20)–(6.21).

Proof

It is an immediate consequence of the commutation relations being satisfied, that the representation (6.9) for $\Psi ^{(\varvec{p}_1)}$ still holds with the definition (6.20)–(6.21). $\square$

The commutation relations impose a simplification on the structure of the matrices $R^{(j)}$. Let the new convention (6.20)–(6.21) be used. The relations $[T^{(i)},R^{(j)}]=0$ for $i=1,\ldots ,p_1$ and $j=q_1+1,\ldots ,p_1$, $j\ne i$, imply the vanishing of the first $p_1$ non-trivial entries of $R^{(j)}$, so that (by (6.11), (6.12) and (6.21)),

$$\begin{aligned} R^{(j)}= & {} \left[ \vec {0}~\left| ~\cdots ~\left| ~\vec {0}~\left| ~ \sum _{m=p_1+1}^n r^{(j)}_m \vec {e}_m ~\right| ~\vec {0}~\right| ~\cdots ~\right| ~\vec {0} \right] ,\quad \quad \lambda ^\prime _j\in \mathbb {N}. \end{aligned}$$

(6.25)

$$\begin{aligned} R^{(j)}= & {} \begin{pmatrix} 0 &{}\cdots &{}&{}&{}&{} \cdots &{} 0 \\ \vdots &{}&{}&{}&{}&{} &{} \vdots \\ 0 &{} \cdots &{}0 &{} 0 &{} r_{p_1+1}^{(j)}&{} \cdots &{} r_n^{(j)} \\ \vdots &{}&{}&{}&{}&{} &{} \vdots \\ 0 &{} \cdots &{}&{}&{}&{} \cdots &{} 0 \end{pmatrix}\quad \longleftarrow \hbox { row } j,\quad \quad \lambda ^\prime _j\in \mathbb {Z}_{-}; \end{aligned}$$

(6.26)

The relations $[R^{(j)},R^{(k)}]=0$ for either $j,k\in \{q_1+1,\dots ,q_1+c_1\}$ or $j,k\in \{q_1+c_1+1,\dots ,p_1\}$ are automatically satisfied. On the other hand, the commutators $[R^{(j)},R^{(k)}]=0$ for $j\in \{q_1+1,\dots ,q_1+c_1\}$ and $k\in \{q_1+c_1+1,\dots ,p_1\}$ imply the further (quadratic) relations

$$\begin{aligned} \sum _{m=p_1+1}^n r^{(j)}_m r^{(k)}_m=0. \end{aligned}$$

(6.27)

In particular, if $\lambda ^\prime _j=-1$ and $R^{(j)}$ is (6.21), all the above conditions can be satisfied, provided that we take $m_j\ge p_1+1$, as we have agreed from the beginning.

6.2 Selected vector solutions $\vec {\Psi }_i$

Remark on notations For the sake of the proof, it is convenient to use a slightly different notation with respect to the statement of Theorem 5.1. The identifications between objects in the proof and objects in the statement is $\vec {\varphi }_i \longmapsto \vec {\psi }_i$, $r_i^{(m)}/r_k^{(i)}\longmapsto r_m$ and $\vec {\varphi }_k/r_k^{(i)} \longmapsto \phi _i$.

We will construct selected vector solutions of Theorem 5.1 from suitable linear combinations of columns of the fundamental matrix $\Psi ^{(\varvec{p}_1)}$ in (6.24). The ith column of an $n\times n$ matrix M is $M\cdot \vec {e}_i$ (rows by columns multiplication), where $\vec {e}_i$ is the standard unit basic vector in $\mathbb {C}^n$. From (6.22)–(6.23), and (6.26)–(6.25)–(6.27), we receive

$$\begin{aligned}&\prod _{l=1}^{p_1} (\lambda -u_l)^{T^{(l)}}\cdot \prod _{j=q_1+1}^{p_1}(\lambda -u_j)^{R^{(j)}}\cdot \vec {e}_i\nonumber \\&\quad = \left\{ \begin{array}{ccc} (\lambda -u_i)^{-\lambda ^\prime _i-1} \vec {e}_i, &{} i=1,\ldots ,q_1+c_1, &{} \lambda ^\prime _i\in \mathbb {C}\backslash \mathbb {N}; \\ (\lambda -u_i)^{-\lambda ^\prime _i-1} \vec {e}_i+\left( \sum _{m=p_1+1}^n r^{(i)}_m \vec {e}_m \right) \ln (\lambda -u_i), &{} i=q_1+c_1+1,\ldots ,p_1,&{} \lambda ^\prime _i\in \mathbb {N}; \\ \vec {e}_i+\sum _{m=q_1+1}^{q_1+c_1} \vec {e}_m r_i^{(m)} (\lambda -u_m)^{-\lambda ^\prime _m-1}\ln (\lambda -u_m), &{} i=p_1+1,\ldots ,n. \end{array} \right. \end{aligned}$$

(6.28)

For $i=1,\ldots ,n$, let

$$\begin{aligned} \vec {\varphi }_i(\lambda ,u):=G^{(\varvec{p}_1)} U(\lambda ,u) \cdot \vec {e}_i,\quad \quad i=1,\ldots ,n, \end{aligned}$$

(6.29)

which is holomorphic for $(\lambda ,u)\in \mathbb {D}_1\times \mathbb {D}(u^c)$. For $i=1,\ldots ,p_1$, we define vector valued functions

$$\begin{aligned} \vec {\Psi }_i(\lambda ,u):= \left\{ \begin{array}{ccc} \vec {\varphi }_i(\lambda ,u) (\lambda -u_i)^{-\lambda ^\prime _i-1},&{} i=1,\ldots ,q_1+c_1,&{} \lambda ^\prime _i\in \mathbb {C}\backslash \mathbb {N}; \\ \sum _{k=p_1+1}^n r_k^{(i)} \vec {\varphi }_k(\lambda ,u), &{} i=q_1+c_1+1,\ldots ,p_1, &{} \lambda ^\prime _i\in \mathbb {N}. \end{array} \right. \end{aligned}$$

(6.30)

Notice that for $i=q_1+c_1+1,\ldots ,p_1$, if $r_k^{(i)}=0$ for all $k=p_1+1,\ldots ,n$, then $\vec {\Psi }_i(\lambda ,u)$ is identically zero

$$\begin{aligned} \vec {\Psi }_i(\lambda ,u)\equiv 0,\quad \lambda ^\prime _i\in \mathbb {N}, \end{aligned}$$

(6.31)

Hence, the ith column of $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$ is

$$\begin{aligned}&\Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_i&=\vec {\Psi }_i(\lambda ,u),&i=1,\ldots ,q_1+c_1, \end{aligned}$$

(6.32)

$$\begin{aligned}&=\vec {\Psi }_i(\lambda ,u)\ln (\lambda -u_i) +\frac{\vec {\varphi }_i(\lambda ,u)}{(\lambda -u_i)^{\lambda ^\prime _i+1}} ,&i=q_1+c_1+1,\ldots ,p_1, \end{aligned}$$

(6.33)

$$\begin{aligned}&=\varphi _i(\lambda ,u)+\sum _{m=q_1+1}^{q_1+c_1} r_i^{(m)}\vec {\Psi }_m(\lambda ,u) \ln (\lambda -u_m),&i=p_1+1,\ldots ,n. \end{aligned}$$

(6.34)

Proposition 6.1

The vector functions (6.30) coincide with the following linear combinations of columns of $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$,

$$\begin{aligned} \vec {\Psi }_i(\lambda ,u)= \left\{ \begin{array}{cc} \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_i, &{} i=1,\ldots ,q_1+c_1, \quad \hbox { namely } \lambda ^\prime _i\in \mathbb {C}\backslash \mathbb {N}; \\ \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \sum _{k=p_1+1}^n r_k^{(i)} \vec {e}_k, &{} i=q_1+c_1+1,\ldots ,p_1, \quad \hbox { namely } \lambda ^\prime _i\in \mathbb {N}. \end{array} \right. \end{aligned}$$

(6.35)

As such, they are vector solutions (called selected) of the Pfaffian system (5.3). Those $\vec {\Psi }_i(\lambda ,u)$ which are not identically zero are linearly independent.

Proof

For $i=1,\ldots ,q_1+c_1$, (6.35) is just (6.32), so it is a vector solution of (5.3). In case $i=q_1+c_1+1,\ldots ,p_1$, we claim that $\vec {\Psi }_i(\lambda ,u)$ defined in (6.30) coincides with the following linear combination

$$\begin{aligned} \vec {\Psi }_i(\lambda ,u)= \sum _{k=p_1+1}^n r_k^{(i)}\left( \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_k\right) , \quad \quad i=q_1+c_1+1,\ldots ,p_1, \end{aligned}$$

of the vector solutions (6.34). Indeed,

$$\begin{aligned}&\sum _{k=p_1+1}^n r_k^{(i)}\left( \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_k\right) \\&\quad = \sum _{k=p_1+1}^n r_k^{(i)}\left( \varphi _k(\lambda ,u)+\sum _{m=q_1+1}^{q_1+c_1} r_k^{(m)}\vec {\Psi }_m(\lambda ,u) \ln (\lambda -u_m)\right) \\&\quad \underset{(6.30)}{=}\vec {\Psi }_i(\lambda ,u) +\sum _{m=q_1+1}^{q_1+c_1}\left( \sum _{k=p_1+1}^n r_k^{(i)}r_k^{(m)}\right) \vec {\Psi }_m(\lambda ,u)\ln (\lambda -u_m). \end{aligned}$$

Now, it follows from (6.27) that $\sum _{k=p_1+1}^n r_k^{(i)}r_k^{(m)}=0$, so proving the claim and the expressions (6.35). Linear independence follows from (6.35). $\square$

6.3 Singular solutions $\vec {\Psi }_i^{(sing)}$

Using the previous results, we define singular vector solutions of the Pfaffian system.

• For $\underline{\lambda ^\prime _i\not \in \mathbb {Z}}$, i.e. $i=1,\ldots ,q_1$,

  $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u):= \vec {\Psi }_i(\lambda ,u)~\equiv \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_i \end{aligned}$$

• For $\underline{\lambda ^\prime _i\in \mathbb {N}}$, i.e. $i=q_1+c_1+1,\ldots ,p_1$,

  $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u):= \vec {\Psi }_i(\lambda ,u)\ln (\lambda -u_i)+\frac{\vec {\varphi }_i(\lambda ,u)}{(\lambda -u_i)^{\lambda ^\prime _i+1}}~\equiv \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_i. \end{aligned}$$

• For $\underline{\lambda ^\prime _i\in \mathbb {Z}_{-}}$, i.e. $i=q_1+1,\ldots ,q_1+c_1$, we distinguish three subcases.

  • i) If $\underline{\lambda ^\prime _i\le -2}$ and $r^{(i)}_k\ne 0$ for some $k\in \{p_1+1,\ldots ,n\}$, from (6.34) (change notation $i\mapsto k$)

    $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u)&:=\frac{1}{r^{(i)}_k}\left\{ \varphi _k(\lambda ,u)+\sum _{m=q_1+1}^{q_1+c_1} r_k^{(m)}\vec {\Psi }_m(\lambda ,u) \ln (\lambda -u_m)\right\} \\&\equiv \frac{1}{r^{(i)}_k} \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_k. \end{aligned}$$

  • ii) If $\underline{\lambda ^\prime _i\le -2}$ and $r^{(i)}_k= 0$ for all $k\in \{p_1+1,\ldots ,n\}$,

    $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u):=0 \end{aligned}$$

  • iii) If $\underline{\lambda ^\prime _i=-1}$, then $r_{m_i}^{(i)}=1$ and in i) above we take $k=m_i$, so that

    $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u)&:=\vec {\varphi }_{m_i}(\lambda ,u)+ \vec {\Psi }_i(\lambda ,u)\ln (\lambda -u_i)\\&\quad +\sum _{m\ne i, ~m=q_1+1}^{q_1+c_1} r_{m_i}^{(m)} \vec {\Psi }_m(\lambda ,u) \ln (\lambda -u_m).\\&=\Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_{m_i},\quad \quad m_i\ge p_1+1. \end{aligned}$$

  The above $\vec {\Psi }_i^{(sing)}(\lambda ,u)$ in i) and iii) is singular at $u_i$, but possibly also at $u_{q_1+1}, \dots , u_{q_1+c_1}$ corresponding to $\lambda ^\prime _m\in \mathbb {Z}_{- }$. By definition,

  $$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u)\underset{\lambda \rightarrow u_i}{=}\vec {\Psi }_i(\lambda ,u) \ln (\lambda -u_i) +\hbox {reg}(\lambda -u_i),\quad \quad i=q_1+1,\ldots ,q_1+c_1, \end{aligned}$$

  (6.36)

Remark 6.3

The definition in (i) contains the freedom of choosing $k\in \{p_1+1,\ldots ,n\}$, which changes $\varphi _k(\lambda ,u)$ and the ratios ${r_k^{(m)}/r^{(i)}_k}$ [in formula (5.8), $\varphi _k/r^{(i)}_k$ is denoted by $\phi _i$ and ${r_k^{(m)}/r^{(i)}_k}$ is $r_m$]. Whatever is the choice of k, provided that $r^{(i)}_k\ne 0$, the behaviour at $\lambda =u_i$ of the corresponding $\vec {\Psi }_i^{(sing)}$ is always (6.36), so it is uniquely fixed if we fix the normalization of $\vec {\Psi }_i(\lambda ,u)$.

As a consequence of the above definitions and Sect. 6.2, we receive the following

Proposition 6.2

The $\vec {\Psi }_i^{(sing)}(\lambda ,u)$ defined above, $i=1,\ldots ,p_1$, when not identically zero, are linearly independent. They are represented as follows

$$\begin{aligned} \vec {\Psi }_i^{(sing)}(\lambda ,u) = \left\{ \begin{array}{ccc} \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \vec {e}_i, &{} \lambda ^\prime _i\in \mathbb {C}\backslash \mathbb {Z}_{-},&{} \\ \Psi ^{(\varvec{p}_1)}(\lambda ,u)\cdot \dfrac{\vec {e}_k}{r_k^{(i)}}, &{} \lambda ^\prime _i\in \mathbb {Z}_{-},&{}\hbox {for some } \in \{p_1+1,\ldots ,n\}\hbox { such that } r_k^{(i)}\ne 0 \\ 0, &{} \lambda ^\prime _i\in -\mathbb {N}-2,&{}\hbox {if } r_k^{(i)}= 0\hbox { for all } k\in \{p_1+1,\ldots ,n\}. \end{array} \right. \end{aligned}$$

6.4 Expansions at $\lambda =u_i$, $i=1,\ldots ,p_1$ and completion of the proof

In order to proceed in the proof, and in view of the Laplace transform to come, we need local behaviour at $\lambda =u_i$.

Lemma 6.3

The following Taylor expansion holds at $\lambda =u_i$, with coefficients $\vec {d}_l^{~(i)}(u)$ holomorphic on $\mathbb {D}(u^c)$,

$$\begin{aligned} \vec {\Psi }_i(\lambda ,u) = \sum _{l=0}^\infty \vec {d}_l^{~(i)}(u) (\lambda -u_i)^l, \quad \quad \lambda ^\prime _i \in \mathbb {N}, ~\hbox { namely }~ i=q_1+c_1+1,\ldots ,p_1. \end{aligned}$$

Proof

By (6.30), $\vec {\Psi }_i(\lambda ,u)= G^{(\varvec{p}_1)} U(\lambda ,u) \cdot (\sum _{m=p_1+1}^n r^{(i)}_m \vec {e}_m )$, so it is holomorphic on $\mathbb {D}_1\times \mathbb {D}(u^c)$. From this we conclude. $\square$

The coefficients $d_l^{(i)}(u)$ will be fixed by a chosen normalization for $\vec {\varphi }_i$ in (6.33), as in the following lemma.

Lemma 6.4

The following Taylor expansions hold at $\lambda =u_i$, uniformly convergent for $u\in \mathbb {D}(u^c)$.

$$\begin{aligned}&\left. \begin{array}{cc} \lambda ^\prime _i\not \in \mathbb {N},\hbox { i.e. } i=1,\ldots ,q_1+c_1: &{} \vec {\Psi }_i(\lambda ,u) \\ \lambda ^\prime _i \in \mathbb {N},\hbox { i.e. } q_1+c_1+1,\ldots ,p_1: &{}\dfrac{\vec {\varphi }_i(\lambda ,u)}{(\lambda -u_i)^{\lambda ^\prime _i+1}} \end{array} \right\} \\&\quad \underset{\lambda \rightarrow u_i}{=} \Bigl (f_i\vec {e}_i+\sum _{l=1}^\infty \vec {b}_l^{~(i)}(u)(\lambda -u_i)^l\Bigr )(\lambda -u_i)^{-\lambda ^\prime _i-1}, \end{aligned}$$

with certain vector coefficients $\vec {b}_l^{~(i)}(u)$ holomorphic in $\mathbb {D}(u^c)$. In particular, the leading term is constant, and will be chosen as follows

$$\begin{aligned} f_i= \left\{ \begin{array}{ccc} \Gamma (\lambda ^\prime _i+1),&{} \lambda ^\prime _i\in \mathbb {C}\backslash \mathbb {Z},&{}i=1,\ldots ,q_1, \\ \dfrac{(-1)^{\lambda ^\prime _i}}{(-\lambda ^\prime _i-1)!}, &{} \lambda ^\prime _i \in \mathbb {Z}_{-},&{} i=q_1+1,\ldots ,q_1+c_1, \\ \lambda ^\prime _i!\equiv \Gamma (\lambda ^\prime _i+1), &{} \lambda ^\prime _i\in \mathbb {N}, &{} i=q_1+c_1+1,\ldots ,p_1. \end{array} \right. \end{aligned}$$

(6.37)

Proof

That the above convergent expansions must hold follows from the definitions. Work is required to prove that the leading term is $f_i\vec {e}_i$, with $f_i\in \mathbb {C}\backslash \{0\}$.

From definitions (6.29)–(6.30), the leading term must coincide with the leading term of the expansion at $\lambda =u_i$ of the ith column $G^{(\varvec{p}_1)} U(\lambda ,u) \cdot \vec {e}_i$, for $i=1,\ldots ,p_1$. To evaluate it, observe that the solution $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$, restricted to a polydisc $\mathbb {D}(u^0)$ contained in a $\tau$-cell of $\mathbb {D}(u^c)$, is a fundamental matrix solution of the Fuchsian system (1.4) in the Levelt form (6.38) at $\lambda =u_i$, $i=1,\ldots ,p_1$. Indeed, by (6.23) it can be written as

$$\begin{aligned} \Psi ^{(\varvec{p}_1)} (\lambda ,u)= & {} \Bigl \{G^{(\varvec{p}_1)} U^{(\varvec{p}_1)} (\lambda ,u) \prod _{ \begin{array}{c} l=1 \\ l\ne i\end{array}}^{p_1} (\lambda -u_l)^{T^{(l)}} \prod _{ \begin{array}{c} j=q_1+1 \\ j\ne i\end{array}}^{p_1}(\lambda -u_j)^{R^{(j)}}\Bigr \}\\&\cdot (\lambda -u_i)^{T^{(i)}} (\lambda -u_i)^{R^{(i)}}, \end{aligned}$$

where it is understood that $R^{(i)}=0$ if $i=1,\ldots ,q_1$. We have

$$\begin{aligned} U^{(\varvec{p}_1)} (\lambda ,u)=I+F_i(u)+O(\lambda -u_i), \quad \quad \lambda \rightarrow u_i, \quad \quad F_{i}(u):=U^{(\varvec{p}_1)} (u_i,u), \end{aligned}$$

and $O(\lambda -u_i)$ represent vanishing terms at $\lambda =u_i$, holomorphic in $\mathbb {D}_1\times \mathbb {D}(u^c)$. The expansion at $\lambda =u_i$ of the factors $(\lambda -u_l)^{T^{(i)}}$ and $(\lambda -u_j)^{R^{(j)}}$, for $l,j\ne i$, yields the Levelt form

$$\begin{aligned} \Psi ^{(\varvec{p}_1)}(\lambda ,u)\underset{\lambda \rightarrow u_i}{=} G^{(i;\varvec{p}_1)}(u)\Bigl (I+O(\lambda -u_i)\Bigr )(\lambda -u_i)^{T^{(i)}} (\lambda -u_i)^{R^{(i)}},\quad i=1,\ldots ,p_1, \end{aligned}$$

(6.38)

where $O(\lambda -u_i)$ are higher order terms, provided that $u\in \mathbb {D}(u^0)$ (they contain negative powers $(u_i-u_k)^{-m}$), and

$$\begin{aligned} G^{(i;\varvec{p}_1)}(u):= G^{(\varvec{p}_1)}(I+F_i(u)) \prod _{ \begin{array}{c} l=1 \\ l\ne i\end{array}}^{p_1} (u_i-u_l)^{T^{(l)}} \prod _{ \begin{array}{c} j=q_1+1 \\ j\ne i\end{array}}^{p_1}(u_i-u_j)^{R^{(j)}},\quad i=1,\ldots ,p_1 . \end{aligned}$$

The matrix $G^{(i;\varvec{p}_1)}(u)$ is holomorphically invertible if restricted to a polydisc $\mathbb {D}(u^0)$ contained in a $\tau$-cell, but it is branched at the coalescence locus $\Delta$ on the whole $\mathbb {D}(u^c)$.

We reach our goal if we show that the ith column $G^{(i;\varvec{p}_1)}(u)\cdot \vec {e}_i$ is constant in $\mathbb {D}(u^c)$. First, it follows from (6.38) and the standard isomonodromic theory of [33] that $G^{(i;\varvec{p}_1)}(u)$ holomorphically in $\mathbb {D}(u^0)$ reduces $B_i(u)$ to the diagonal form $T^{(i)}$, when $\lambda ^\prime _i\ne -1$,

$$\begin{aligned} \Bigl (G^{(i;\varvec{p}_1)}(u)\Bigr )^{-1} B_i(u) ~G^{(i;\varvec{p}_1)}(u)=T^{(i)}, \end{aligned}$$

or to non-diagonal Jordan form (6.21) when $\lambda ^\prime _i =-1$

$$\begin{aligned} \Bigl (G^{(i;\varvec{p}_1)}(u)\Bigr )^{-1} B_i(u) ~G^{(i;\varvec{p}_1)}(u)=R^{(i)}\equiv J^{(i)},\quad \quad \lambda ^\prime _i =-1. \end{aligned}$$

For this reason, the ith row is proportional to the eigenvector $\vec {e}_i$ of $B_i(u)$ relative to the eigenvalue $-\lambda ^\prime _i-1$. Namely, for some scalar function $f_i(u)$,

$$\begin{aligned} G^{(i;\varvec{p}_1)}(u)\cdot \vec {e}_i= f_i(u) \vec {e}_i. \end{aligned}$$

This is obvious for $\lambda ^\prime _i\ne -1$, namely for diagonalizable $B_i$. If $\lambda ^\prime _i=-1$, the eigenvalue 0 of $B_i$ appearing in $J^{(i)}$ at entry (i, i) is associated with the eigenvector $f_i(u) \vec {e}_i$. Moreover, for every invertible matrix $G=[*|\cdots |*|\vec {e}_i|*|\cdots |*]$, where $\vec {e}_i$ occupies the kth column, then $G^{-1}B_i(u) G$ is zero everywhere, except for the kth row. Now, since $R^{(i)}=J^{(i)}$ has only one non-zero entry on the ith row, it follows that the eigenvector $f_i(u) \vec {e}_i$ must occupy the ith column of $G^{(i;\varvec{p}_1)}(u)$.

• $f_i(u)$ is holomorphic on $\mathbb {D}(u^c)$. Indeed, by (6.28),

  $$\begin{aligned} \prod _{ \begin{array}{c} l=1 \\ l\ne i\end{array}}^{p_1} (u_i-u_l)^{T^{(l)}} \prod _{ \begin{array}{c} j=q_1+1 \\ j\ne i\end{array}}^{p_1}(u_i-u_j)^{R^{(j)}}\cdot \vec {e}_i=\vec {e}_i. \end{aligned}$$

  Therefore $f_i(u) \vec {e}_i \equiv G^{(i;\varvec{p}_1)}(u)\cdot \vec {e}_i ~ = G^{(\varvec{p}_1)}(I+F_i(u))\vec {e}_i$. We conclude, because $F_i(u)$ is holomorphic on $\mathbb {D}(u^c)$.

• $f_i$ is constant on $\mathbb {D}(u^c)$. Indeed, since $\Psi ^{(\varvec{p}_1)}(\lambda ,u)$ is an isomonodromic solution in $\mathbb {D}(u^0)$, the matrix $G^{(i;\varvec{p}_1)}(u)$ must satisfy the Pfaffian system (see Appendix A, identify $G^{(i;\varvec{p}_1)}$ with $G^{(i)}$ in Corollary 9.1)

  $$\begin{aligned} \dfrac{\partial G^{(i;\varvec{p}_1)}}{\partial u_j}= & {} \left( \dfrac{B_j}{u_j-u_i} +\omega _j\right) G^{(i;\varvec{p}_1)}, \quad j\ne i ; \quad \quad \dfrac{\partial G^{(i;\varvec{p}_1)}}{\partial u_i}\nonumber \\= & {} \sum _{j\ne i} \left( \dfrac{B_j}{u_i-u_j}+\omega _j\right) G^{(i;\varvec{p}_1)} . \end{aligned}$$

  (6.39)

  From (2.18) and (4.3), the ith column of $\dfrac{B_j}{u_j-u_i} +\omega _j$ is null. Hence,

  $$\begin{aligned} \frac{\partial }{\partial u_j}\left( G^{(i;\varvec{p}_1)}\cdot \vec {e}_i\right) =0, \quad \forall j\ne i . \end{aligned}$$

  Moreover, summing the Eq. (6.39), we get $\sum _{j=1}^n \partial _j G^{(i;\varvec{p}_1)}=0$. Thus, $G^{(i;\varvec{p}_1)}\cdot \vec {e}_i$ is constant on $\mathbb {D}(u^0)$, and being holomorphic on $\mathbb {D}(u^c)$, it is constant on $\mathbb {D}(u^c)$. The choice (6.37) will be made. $\square$

The above obtained expansions for the $\vec {\Psi }_i$ and $\vec {\Psi }_i^{(sing)}$ and $\vec {\varphi }_i$ prove Theorem 5.1 for $i=1,\ldots ,p_1$, with some obvious identifications between objects in the proof and objects in the statement, namely $\vec {\varphi }_i \longmapsto \vec {\psi }_i$, $r_i^{(m)}/r_k^{(i)}\longmapsto r_m$ and $\vec {\varphi }_k/r_k^{(i)} \longmapsto \phi _i$.

6.5 Analogous proof for all coalescences

With the labelling (6.1)–(6.2), the same strategy above holds for every coalescence

$$\begin{aligned} (u_{p_1+\cdots +p_{\alpha -1}+1}, \ldots , u_{p_1+\cdots +p_{\alpha }})\longrightarrow (\lambda _\alpha ,\ldots ,\lambda _\alpha ), \quad \alpha =1,\ldots ,s. \end{aligned}$$

We find corresponding isomonodromic fundamental matrices for the Pfaffian system (with self-explaining notations)

$$\begin{aligned} \Psi ^{(\varvec{p}_\alpha )}(\lambda ,u)= & {} G^{(\varvec{p}_\alpha )}\cdot U^{(\varvec{p}_\alpha )}(\lambda ,u)\cdot \prod _{l=p_1+\cdots +p_{\alpha -1}+1}^{p_1+\cdots +p_{\alpha }} (\lambda -u_l)^{T^{(l)}}\\&\quad \prod _{j=(p_1+\cdots +p_{\alpha -1}+1)+q_\alpha }^{p_1+\cdots +p_{\alpha }}(\lambda -u_j)^{R^{(j)}}. \end{aligned}$$

where $\varvec{p}_\alpha =(p_1+\cdots +p_{\alpha -1}+1,\dots ,p_1+\cdots +p_{\alpha })$. Then, we proceed in the same way, constructing the solutions $\vec {\Psi }_i$ and $\vec {\Psi }_i^{(sing)}$, with $p_1+\cdots +p_{\alpha -1}+1\le i \le p_1+\cdots +p_{\alpha }$. $\square$

6.6 Proof of Proposition 5.1

Proof

For simplicity, we omit $\nu$ in the connection coefficients $c_{jk}^{(\nu )}$ in (5.12)–(5.13). It follows from the very definitions of the $\vec {\Psi }_k$ and $\vec {\Psi }_j^{(sing)}$ that

$$\begin{aligned} c_{jk}=0\quad \hbox { if } u_j^c=u_k^c. \end{aligned}$$

In order to prove independence of u, we express the monodromy of

$$\begin{aligned} \Psi (\lambda ,u):=[\vec {\Psi }_1(\lambda ,u)~|~\cdots ~|\vec {\Psi }_n(\lambda ,u)], \end{aligned}$$

in terms of the connection coefficients. From the definition, we have (using the notations in the statement of Theorem 5.1)

$$\begin{aligned} \vec {\Psi }_k(\lambda ,u) = \left\{ \begin{array}{ll} \vec {\Psi }_j(\lambda ,u)c_{jk}+\hbox {reg}(\lambda -u_j),&{} \lambda ^\prime _j\not \in \mathbb { Z} \\ \\ \vec {\Psi }_j(\lambda ,u)\ln (\lambda -u_j)c_{jk}+\hbox {reg}(\lambda -u_j),&{} \lambda ^\prime _j\in \mathbb {Z}_{-} \\ \\ \left( \vec {\Psi }_j(\lambda ,u)\ln (\lambda -u_j)+\dfrac{\psi _j(\lambda ,u)}{ (\lambda -u_j)^{\lambda ^\prime _j+1}}\right) c_{jk}+\hbox {reg}(\lambda -u_j),&{} \lambda ^\prime _j\in \mathbb { N} \end{array} \right. \end{aligned}$$

(6.40)

For $u\not \in \Delta$ and a small loop $(\lambda -u_k)\mapsto (\lambda -u_k)e^{2\pi i}$ we obtain from Theorem 5.1

$$\begin{aligned} \vec {\Psi }_k(\lambda ,u) \longmapsto \vec {\Psi }_k(\lambda ,u) e^{-2\pi i \lambda ^\prime _k}, \quad \hbox { which includes also the case } \lambda ^\prime _k\in \mathbb {Z}, \hbox { with } e^{-2\pi i \lambda ^\prime _k}=1. \end{aligned}$$

For a small loop $(\lambda -u_j)\mapsto (\lambda -u_j)e^{2\pi i}$, $j\ne k$, from Theorem 5.1 and (6.40) we obtain

$$\begin{aligned}&\vec {\Psi }_k \longmapsto \vec {\Psi }_j e^{-2\pi i \lambda ^\prime _j} c_{jk} +\underbrace{\hbox {reg}(\lambda -u_j)}_{\vec {\Psi }_k-\vec {\Psi }_jc_{jk}}=\vec {\Psi }_k+(e^{-2\pi i \lambda ^\prime _j}-1)c_{jk} \vec {\Psi }_j&\hbox {for } \lambda ^\prime _j\not \in \mathbb {Z} \\&\vec {\Psi }_k \longmapsto \vec {\Psi }_j\Bigl (\ln (\lambda -u_j)+2\pi i\Bigr )c_{jk}+\hbox {reg}(\lambda -u_j)= \vec {\Psi }_k+2\pi i c_{jk} \vec {\Psi }_j,&\hbox {for } \lambda ^\prime _j\in \mathbb {Z}_{-} \\&\vec {\Psi }_k \longmapsto \left( \vec {\Psi }_j\Bigl (\ln (\lambda -u_j)+2\pi i\Bigr )+\dfrac{\psi _j(\lambda ,u)}{ (\lambda -u_j)^{\lambda ^\prime _j+1}}\right) c_{jk}+\hbox {reg}(\lambda -u_j)= \vec {\Psi }_k+ 2\pi i c_{jk} \vec {\Psi }_j,&\hbox { for } \lambda ^\prime _j\in \mathbb {N}. \end{aligned}$$

Therefore, for $u\not \in \Delta$ and a small loop $\gamma _k:(\lambda -u_k)\mapsto (\lambda -u_k)e^{2\pi i}$ not encircling other points $u_j$ (we denote the loop by $\lambda \mapsto \gamma _k\lambda$), we receive

$$\begin{aligned} \Psi (\lambda ,u)\longmapsto \Psi (\gamma _k\lambda ,u)=\Psi (\lambda ,u) M_k(u), \end{aligned}$$

where

$$\begin{aligned}&(M_k)_{jj}=1\quad j\ne k,\quad (M_k)_{kk}=e^{-2\pi i \lambda ^\prime _k};\quad \quad (M_k)_{kj}= \alpha _kc_{kj},\quad j\ne k;\\&\quad (M_k)_{ij}=0 \hbox { otherwise}. \end{aligned}$$

and

$$\begin{aligned} \alpha _k:=(e^{-2\pi i \lambda ^\prime _k}-1), \quad \hbox { if }\lambda ^\prime _k\not \in \mathbb {Z};\quad \quad \quad \alpha _k:=2\pi i ,\quad \hbox { if }\lambda ^\prime _k\in \mathbb {Z}. \end{aligned}$$

We proceed by first analyzing the generic case, and then the general case.

Generic case. Suppose that A(u) has no integer eigenvalues (recall that eigenvalues do not depend on u). Let us fix u in a $\tau$-cell. By Proposition 2.4, $\Psi (\lambda ,u)$ is a fundamental matrix solution of (1.4) for the fixed u, and $C=(c_{jk})$ is invertible. Thus

$$\begin{aligned} M_k(u)= \Psi (\gamma _k\lambda ,u)\Psi (\lambda ,u)^{-1}. \end{aligned}$$

The above makes sense for every u in the considered $\tau$-cell, being $\Psi (\lambda ,u)$ invertible at such an u. But $\Psi (\lambda ,u)$ and $\Psi (\gamma _k\lambda ,u)$ are holomorphic on $\mathcal {P}_\eta (u)\hat{\times } \mathbb {D}(u^c)$, so that the matrix $M_k(u)$ is holomorphic on the $\tau$-cell. Repeating the above argument for another $\tau$-cell, we conclude that $M_k(u)$ is holomorphic on each $\tau$-cell. Now, on a $\tau$-cell, we have

$$\begin{aligned} \mathrm{d}\Psi (\gamma _k\lambda ,u)=P(\lambda ,u)\Psi (\gamma _k\lambda ,u)=P(\lambda ,u)\Psi (\lambda ,u) M_k, \end{aligned}$$

and at the same time

$$\begin{aligned} \mathrm{d}\Psi (\gamma _k\lambda ,u)= & {} \mathrm{d}\Bigl (\Psi (\lambda ,u) M_k\Bigr )= \mathrm{d}\Psi (\lambda ,u)~M_k+\Psi (\lambda ,u)~dM_k\\= & {} P(\lambda ,u)\Psi (\lambda ,u) M_k+\Psi (\lambda ,u)~dM_k. \end{aligned}$$

The two expressions are equal if and only if $dM_k=0$, because $\Psi (\lambda ,u)$ is invertible on a $\tau$-cell. Recall that $\tau$-cells are disconnected from each other, so that separately on each cell, $M_k$ is constant, and so the connection coefficients are constant separately on each cell.

We further suppose that none of the $\lambda ^\prime _j$ is integer. In this case, $\vec {\Psi }^{(sing)}_j=\vec {\Psi }_j$ for all $j=1,\ldots ,n$, so that from (6.40) for $u_k^c\ne u_j^c$ (otherwise $c_{jk}=0$ and there is nothing to prove)

$$\begin{aligned} \vec {\Psi }_k(\lambda ,u)\underset{\lambda \rightarrow u_j}{=} \vec {\Psi }_j(\lambda ,u)c_{jk}+ \hbox {reg}(\lambda -u_j). \end{aligned}$$

(6.41)

Using the labelling (6.1)–(6.2), from the Proof of Theorem 6.1 we have the fundamental matrix solution

$$\begin{aligned} \Psi ^{(\varvec{p}_1)}(\lambda ,u)= \Bigl [ \vec {\Psi }_1(\lambda ,u)~\Bigr |~\cdots ~\Bigr |\vec {\Psi }_{p_1}(\lambda ,u)~\Bigr |~\vec {\varphi }_{p_1+1}^{~(1)}(\lambda ,u)~\Bigr |~\cdots ~\Bigr |~\vec {\varphi }_{n}^{~(1)}(\lambda ,u) \Bigr ] \end{aligned}$$

and in general at each $\lambda _\alpha$, $\alpha =1,\ldots ,s$ (with $\sum _{j=1}^{\alpha -1}p_j=0$ for $\alpha =1$) we have

$$\begin{aligned}&\Psi ^{(\varvec{p}_\alpha )}(\lambda ,u) \\&\quad = \Bigl [ \vec {\varphi }_{1}^{~(\alpha )}(\lambda ,u)~\Bigl |~\cdots ~\Bigl |~ \vec {\varphi }_{\sum _{j=1}^{\alpha -1}p_j}^{~(\alpha )}(\lambda ,u) ~\Bigl | \vec {\Psi }_{\sum _{j=1}^{\alpha -1}p_j+1}(\lambda ,u)~\Bigl |~ \vec {\Psi }_{\sum _{j=1}^{\alpha -1}p_j+2}(\lambda ,u)~\Bigl |\\&\qquad \cdots ~\Bigr |\vec {\Psi }_{\sum _{j=1}^{\alpha }p_j}(\lambda ,u)~\Bigr | \\&\qquad \Bigr |~\vec {\varphi }_{\sum _{j=1}^{\alpha }p_j+1}^{~(\alpha )}(\lambda ,u)~\Bigr |~\cdots ~|~\vec {\varphi }_{n}^{~(\alpha )}(\lambda ,u) \Bigr ] \end{aligned}$$

where

$$\begin{aligned} \vec {\Psi }_m(\lambda ,u)=\vec {\psi }_m(\lambda ,u)(\lambda -u_m)^{-\lambda ^\prime _m-1}, \quad m=\sum _{j=1}^{\alpha -1}p_j+1,~ \dots ~,\sum _{j=1}^{\alpha }p_j, \end{aligned}$$

and the $\vec {\psi }_m(\lambda ,u)$ and $\vec {\varphi }_{r}^{~(\alpha )}(\lambda ,u)$ are holomorphic functions in the corresponding $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$. The above allows us to explicitly rewrite (6.41), for j such that $u_j^c=\lambda _\alpha$, as

$$\begin{aligned} \vec {\Psi }_k(\lambda ,u)= \sum _{m=p_{1}+\cdots +p_{\alpha -1}+1}^{p_{1}+\cdots +p_{\alpha }} c_{mk}~ \vec {\Psi }_m(\lambda ,u)+ \sum _{r\not \in \{p_{1}+\cdots +p_{\alpha -1}+1,\ldots ,p_{1}+\cdots +p_{\alpha }\}} h_r \vec {\varphi }_r^{~(\alpha )}(\lambda ,u), \end{aligned}$$

(6.42)

for suitable constant coefficients $h_r$. Here one of the $c_{mk}$ is $c_{jk}$ of (6.41).

Each $u_m$, with $m=p_{1}+\cdots +p_{\alpha -1}+1, \ldots , p_{1}+\cdots +p_{\alpha }$, varies in $\mathbb {D}_\alpha$. Firstly, we can fix $\lambda =\lambda _\alpha$ in (6.42), consider the branch cut $\mathcal {L}_\alpha$ from $\lambda _\alpha$ to infinity in direction $\eta$ (see Fig. 3), and let u vary in such a way that each $u_{p_{1}+\cdots +p_{\alpha -1}+1}$, ..., $u_{p_{1}+\cdots +p_{\alpha }}$ varies in $\mathbb {D}_\alpha \backslash \mathcal {L}_\alpha$, so that in the r.h.s. of (6.42) all the $\vec {\Psi }_m(\lambda _\alpha ,u)$ and $\vec {\varphi }_r^{~(\alpha )}(\lambda _\alpha ,u)$ are holomorphic with respect to u, provided that $u_m\ne \lambda _\alpha$. If u varies, with the constraint that the $u_m$’s must remain in $\mathbb {D}_\alpha \backslash \mathcal {L}_\alpha$, every $\tau$-cell of $\mathbb {D}(u^c)$ can be reached starting from an initial point in one specific cell. This proves, by u-analytic continuation of (6.42) with fixed $\lambda =\lambda _\alpha$, that the coefficients $c_{mk}$ are constant¹⁶ in .

Now, we can slightly vary $\eta$ in $\eta _{\nu +1}<\eta <\eta _\nu$, so that the cut $\mathcal {L}_\alpha$ is irrelevant¹⁷. Thus, the $c_{mk}$ are constant on $\left\{ u\in \mathbb {D}(u^c) ~|~ u_{p_1+\cdots +p_{\alpha -1}+1}\ne \lambda _\alpha ,\dots , u_{p_1+\cdots +p_{\alpha }}\ne \lambda _\alpha \right\}$.

Finally, we fix another value $\lambda =\lambda ^*\in \mathbb {D}_\alpha$ in (6.42), and repeat the above discussion with cuts $\mathcal {L}_\alpha$ issuing from $\lambda ^*$, so that all the $c_{mk}$ are constant on $\left\{ u\in \mathbb {D}(u^c) ~|~ u_{p_1+\cdots +p_{\alpha -1}+1}\ne \lambda ^*,\dots , u_{p_1+\cdots +p_{\alpha }}\ne \lambda ^*\right\}$. This proves constancy of the $c_{mk}$, m associated with $\lambda _\alpha$, on the whole $\mathbb {D}(u^c)$. Then, we repeat this for all $\alpha =1,..,s$, proving constancy of the $c_{jk}$ for all $j=1,\ldots ,n$. Hence, Proposition 5.1 is proved in the generic case.

General case of any A(u). If some of the diagonal entries $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n$ of A are integers, or some eigenvalues are integers, there exists a sufficiently small $\gamma _0>0$ such that, for any $0<\gamma <\gamma _0$, $A-\gamma I$ has diagonal non-integer entries $\lambda ^\prime _1-\gamma ,\ldots ,\lambda ^\prime _n-\gamma$ and no integer eigenvalues. Take such a $\gamma _0$, and for any $0<\gamma <\gamma _0$ consider

$$\begin{aligned} (\Lambda -\lambda )\frac{\mathrm{d}}{ \mathrm{d}\lambda } (~{}_\gamma \Psi )= \Bigl ((A(u)-\gamma I) +I\Bigr )~{}_\gamma \Psi . \end{aligned}$$

(6.43)

namely

$$\begin{aligned} \frac{\mathrm{d}}{ \mathrm{d}\lambda }({}_\gamma \Psi )=\sum _{k=1}^n \frac{B_k[\gamma ](u) }{ \lambda -u_k}{}_\gamma \Psi ,~~~~~B_k[\gamma ](u):=-E_k\Bigl (A(u)+(1-\gamma )I\Bigr ). \end{aligned}$$

(6.44)

Lemma 6.5

The above system (6.44) is strongly isomonodromic in $\mathbb {D}(u^0)$ contained in a $\tau$-cell, and $\lambda$-component of the integrable Pfaffian system

$$\begin{aligned} d{}_\gamma \Psi =P_{[\gamma ]}(\lambda ,u) {}_\gamma \Psi , \quad \quad P_{[\gamma ]}(\lambda ,u) = \sum _{k=1}^n \frac{B_k[\gamma ](u)}{\lambda -u_k} d(\lambda -u_k) +\sum _{j=1}^n [F_1(u),E_j]\mathrm{d}u_j. \end{aligned}$$

(6.45)

where $F_1(u)$ is defined as in (2.8), $(F_1)_{i j}=\frac{A_{ij}}{u_j-u_i}$, $i\ne j$, and $[F_1(u),E_j]$ is (2.18).

Proof

We do a gauge transformation

$$\begin{aligned} {}_\gamma Y(z):=z^{-\gamma } Y(z),\quad \quad \gamma \in \mathbb {C}, \end{aligned}$$

(6.46)

which transforms (1.1) into

$$\begin{aligned} \frac{\mathrm{d}({}_\gamma Y)}{\mathrm{d}z}=\left( \Lambda +\frac{A-\gamma I}{ z} \right) ~ {}_\gamma Y \end{aligned}$$

(6.47)

For $u\in \mathbb {D}(u^0)$ contained in a $\tau$-cell, we write the unique formal solution

$$\begin{aligned} {}_\gamma Y_F(z,u)=z^{-\gamma }Y_F(z,u), \end{aligned}$$

(6.48)

where $Y_F(z,u)$ is (2.4), so that

$$\begin{aligned} {}_\gamma Y_F(z,u)=F(z,u) z^{B-\gamma I} e^{\Lambda z},\quad \quad B-\gamma I=\hbox {diag}(A-\gamma )= \mathrm{diag}(\lambda ^\prime _1-\gamma ,~\ldots ~,\lambda ^\prime _n-\gamma ). \end{aligned}$$

The crucial point is that F(z, u) is the same as (2.5), so all the $F_k(u)$ are independent of $\gamma$. The fundamental matrix solutions

$$\begin{aligned} {}_\gamma Y_\nu (z,u):=z^{-\gamma } Y_\nu (z,u), \end{aligned}$$

are uniquely defined by their asymptotics ${}_\gamma Y_F(z,u)$ in $\mathcal {S}_\nu (\mathbb {D}(u^0))$. Their Stokes matrices do not depend on $\gamma$ because

$$\begin{aligned} {}_\gamma Y_{\nu +(k+1)\mu }(z,u)= {}_\gamma Y_{\nu +k\mu }(z,u) \mathbb {S}_{\nu +k\mu }\quad \Longleftrightarrow \quad Y_{\nu +(k+1)\mu }(z,u)= Y_{\nu +k\mu }(z,u) \mathbb {S}_{\nu +k\mu }. \end{aligned}$$

The system (6.47) is thus strongly isomonodromic. By Proposition 3.1 we conclude. $\square$

Corollary 6.2

Let the assumptions of Theorem 5.1 hold. Then Theorem 5.1 holds also for (6.45).

By Theorem 5.1 applied to (6.45), we receive independent vector solutions ${}_\gamma \vec {\Psi }_k(\lambda ,u)\equiv {}_\gamma \vec {\Psi }^{(sing)}_k(\lambda ,u)$, $k=1,\ldots ,n$, which form a fundamental matrix

$$\begin{aligned} {}_\gamma \Psi (\lambda ,u):=[{}_\gamma \vec {\Psi }_1(\lambda ,u)~|~\cdots ~|{}_\gamma \vec {\Psi }_n(\lambda ,u)]. \end{aligned}$$

For system (6.45) the results already proved in the generic case hold. Therefore, the connection coefficients $c_{jk}^{(\nu )}[\gamma ]$ defined by

$$\begin{aligned} {}_\gamma \vec {\Psi }_k(\lambda ,u~|\nu )={}_\gamma \vec {\Psi }_j(\lambda ,u~|\nu ) ~c_{jk}^{(\nu )}[\gamma ]~+\hbox {reg}(\lambda -u_j),\quad \quad \lambda \in \mathcal {P}_\eta , \end{aligned}$$

(6.49)

are constant on $\mathbb {D}(u^c)$. They depend on $\gamma$, but not on $u\in \mathbb {D}(u^c)$.

Remark 6.4

It is explained in section 8 of [23] what is the relation between $\vec {\Psi }^{(sing)}_k$ and ${}_\gamma \vec {\Psi }_k$, by means of their primitives, and that in general both $\lim _{\gamma \rightarrow 0} {}_\gamma \vec {\Psi }_k$ and $\lim _{\gamma \rightarrow 0}c_{jk}^{(\nu )}[\gamma ]$ are divergent.

Now, we invoke Proposition 10 of [23], which holds with no assumptions on eigenvalues and diagonal entries of A(u).¹⁸ This result, adapted to our case, reads as follows.

Proposition 6.3

Let u be fixed in a $\tau$-cell. Let $\gamma _0>0$ be small enough such that for any $0<\gamma <\gamma _0$ the matrix $A-\gamma I$ has no integer eigenvalues, and its diagonal part has no integer entries.¹⁹ Let $c_{jk}^{(\nu )}$ be the connection coefficients of the Fuchsian system (1.4) at the fixed u, as in Definition 5.1. Let $c_{jk}^{(\nu )}[\gamma ]$ be the connection coefficients in (6.49). Let

$$\begin{aligned} \alpha _k:=\left\{ \begin{array}{cc} e^{-2\pi i \lambda ^\prime _k}-1,&{}\quad \lambda ^\prime _k\not \in \mathbb {Z} \\ 2\pi i, &{} \quad \lambda ^\prime _k\in \mathbb {Z} \end{array} \right. ; ~~~~~~~~\alpha _k[\gamma ]:=e^{-2\pi i (\lambda ^\prime _k-\gamma )}-1 \end{aligned}$$

Then, the following equalities hold

$$\begin{aligned} \alpha _k c_{jk}^{(\nu )}= e^{-2\pi i \gamma }\alpha _k[\gamma ]~c_{jk}^{(\nu )}[\gamma ],\quad \hbox { if }k\succ j;\quad \quad \alpha _k c_{jk}^{(\nu )}= \alpha _k[\gamma ]~c_{jk}^{(\nu )}[\gamma ] , \quad \hbox { if } k\prec j; \end{aligned}$$

(6.50)

where the ordering relation $j \prec k$ means, for the fixed u, that $\mathfrak {R}(z(u_j-u_k))<0$ for $\arg z=\tau =3\pi /2-\eta$ satisfying (5.2).

We use Proposition 6.3 to conclude the proof of Proposition 5.1 in the general case. Indeed, the proposition is already proved in the generic case, so it holds for the $c_{jk}^{(\nu )}[\gamma ]$. Therefore, they are constant on the whole $\mathbb {D}(u^c)$. Equalities (6.50) hold at any fixed u in $\tau$-cell, so that each $c_{jk}^{(\nu )}$ is constant on a $\tau$-cell, and such constant is the same in each $\tau$-cell. With a slight variation of $\eta$ in $(\eta _{\nu +1},\eta _\nu )$, equalities (6.50) hold also at the crossing locus $X(\tau )$. They analytically extend at $\Delta$. $\square$

7 Laplace transform in $\mathbb {D}(u^c)$, Theorem 7.1

By means of the Laplace transform with deformation parameters, we prove points (I1),(I2), (I3), (II1), (II2) and (II5) of Theorem 2.2. Stokes matrices will be expressed in terms of the isomonodromic connection coefficients satisfying Proposition 5.1. The result is in Theorem 7.1, which is the last step of our construction.

Let $\tau$ be the chosen direction in the z-plane admissible at $u^c$, and $\eta =3\pi /2-\tau$ in the $\lambda$-plane. The Stokes rays of $\Lambda (u^c)$ will be labelled as in (2.21), so that (5.2) holds for a certain $\nu \in \mathbb {Z}$. We define the sectors

$$\begin{aligned} \mathcal {S}_\nu =\{z\in \mathcal {R}(\mathbb {C}\backslash \{0\})~\hbox { such that }~\tau _\nu -\pi<\arg z <\tau _{\nu +1}\}. \end{aligned}$$

(7.1)

If u only varies in $\mathbb {D}(u^0)$ contained in a ${\tau }$-cell, then none of the Stokes rays associated with $\Lambda (u)$ crosses $\arg z={\tau }$ mod $\pi$. If u varies in $\mathbb {D}(u^c)$, some Stokes rays associated with $\Lambda (u)$ necessarily cross $\arg z={\tau }$ mod $\pi$ (see Sect. 2.1.2). Consider the subset of the set of Stokes rays satisfying $\mathfrak {R}(z(u_j-u_k))=0$, $z\in \mathcal {R}$, associated with pairs $(u_j,u_k)$ such that $u_j\in \mathbb {D}_\alpha$ and $u_k\in \mathbb {D}_\beta$, $\alpha \ne \beta$, namely $u_j^c\ne u_k^c$. Following [13], we denote this subset by $\mathfrak {R}(u)$. If u varies in $\mathbb {D}(u^c)$ and $\epsilon _0$ satisfies (5.1), the rays in $\mathfrak {R}(u)$ continuously rotate, but never cross the admissible rays $\arg z={\tau }+h\pi$, where

$$\begin{aligned} \tau _{\nu +h\mu }<{\tau }+h\pi <\tau _{\nu +h\mu +1}, \quad \quad h\in \mathbb {Z}, \end{aligned}$$

(7.2)

The above allows to define $\widehat{\mathcal {S}}_{\nu +h\mu }(u)$ to be the unique sector containing $S\bigl ({\tau }+(h-1)\pi ,{\tau }+h\pi \bigr )$ and extending up to the nearest Stokes rays in $\mathfrak {R}(u)$. Then, let

$$\begin{aligned} \widehat{\mathcal {S}}_{\nu +h\mu }:=\bigcap _{u\in \mathbb {D}(u^c)} \widehat{\mathcal {S}}_{\nu +h\mu }(u). \end{aligned}$$

(7.3)

It has angular amplitude greater than $\pi$. The reason for the labeling is that $\widehat{\mathcal {S}}_{\nu +h\mu }(u^c)=\mathcal {S}_{\nu +h\mu }$ in (7.1).

Suppose that u is fixed in a $\tau$-cell. Let

$$\begin{aligned} Y_{\nu +h\mu }(z,u):=\Bigl [\vec {Y}_1(z,u~|\nu +h\mu )~\Bigr | ~\dots ~\Bigr | ~ \vec {Y}_n(z,u~|\nu +h\mu )\Bigr ], \end{aligned}$$

be defined by

$$\begin{aligned}&\vec {Y}_k(z,u~|\nu +h\mu ):= \dfrac{1}{2\pi i } \int _{\gamma _k(\eta -h\pi )} e^{z\lambda } \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu +h\mu ) \mathrm{d}\lambda ,&\hbox { for } \lambda ^\prime _k\not \in \mathbb {Z}_{-}, \end{aligned}$$

(7.4)

$$\begin{aligned}&\vec {Y}_k(z,u~|\nu +h\mu ):= \int _{L_k(\eta -h\pi )} e^{z\lambda } \vec {\Psi }_k(\lambda ,u~|\nu +h\mu ) \mathrm{d}\lambda ,&\hbox { for } \lambda ^\prime _k\in \mathbb {Z}_{-}. \end{aligned}$$

(7.5)

In the $\lambda$-plane, the admissible directions $\eta -h\pi$ correspond to ${\tau }+h\pi$, with

$$\begin{aligned} \eta _{\nu +h\mu +1}<\eta -h\pi < \eta _{\nu +h\mu }. \end{aligned}$$

(7.6)

Here, $\vec {\Psi }_k(\lambda ,u~|\nu +h\mu )$, $\vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu +h\mu )$ are the vector solutions of Theorem 5.1 for $\lambda \in \mathcal {P}_{\eta -h\pi }(u)$, with u fixed in a $\tau$-cell. $L_k(\eta -h\pi )$ is the cut in direction $\eta -h\pi$, issuing from $u_k$ and oriented from $u_k$ to $\infty$, and $\gamma _k(\eta -h\pi )$ is the path coming from $\infty$ along the left side of $L_k(\eta -h\pi )$, encircling $u_k$ with a small loop excluding all the other poles, and going back to $\infty$ along the right side of $L_k(\eta -h\pi )$. Here “right” and “left” refer to the orientation of $L_k(\eta -h\pi )$. The label $\nu +h\mu$ keeps track of (5.2) and (7.2)–(7.6).

Theorem 7.1

Let the assumptions of Theorem 5.1 hold.

(1):

The matrices $Y_{\nu +h\mu }(z,u)$, obtained by Laplace transform (7.4)–(7.5) at a fixed $u\in \mathbb {D}(u^0)$ contained in a $\tau$-cell, define holomorphic matrix valued functions of $(\lambda ,u)\in \mathcal {R}(\mathbb {C}\backslash \{0\})\times \mathbb {D}(u^c)$, which are fundamental matrix solutions of (1.1).

(2):

They have structure

$$\begin{aligned} Y_{\nu +h\mu }(z,u)=\widehat{Y}_{\nu +h\mu }(z,u) z^{B} e^{z\Lambda },\quad \quad B=\hbox {diag}(\lambda ^\prime _1,\ldots ,\lambda ^\prime _n), \end{aligned}$$

with asymptotic behaviour, uniform in $u\in \mathbb {D}(u^c)$,

$$\begin{aligned} \widehat{Y}_{\nu +h\mu }(z,u) \sim F(z,u)=I+\sum _{l=1}^\infty \frac{F_l(u)}{z^l}, \quad \quad z\rightarrow \infty \hbox { in } \widehat{S}_{\nu +h\mu }, \end{aligned}$$

given by the formal solution $Y_F(z,u)=F(z,u)z^{B} e^{z\Lambda }$. The coefficients $F_l(u)$ are holomorphic in $\mathbb {D}(u^c)$. The explicit expression of their columns is (7.12), (7.13), (7.15) [or (7.16)] and (7.17).

(3):

Stokes matrices defined by

$$\begin{aligned} Y_{\nu +(h+1)\mu }(z,u)=Y_{\nu +h\mu }(z,u) \mathbb {S}_{\nu +h\mu }, \quad \quad z\in \widehat{\mathcal {S}}_{\nu +h\mu }\cap \widehat{\mathcal {S}}_{\nu +(h+1)\mu }, \end{aligned}$$

(7.7)

are constant in the whole $\mathbb {D}(u^c)$ and satisfy

$$\begin{aligned} (\mathbb {S}_{\nu +h\mu })_{ab}=(\mathbb {S}_{\nu +h\mu })_{ba}=0 \quad \hbox { for } a\ne b\hbox { such that } u_a^c=u_b^c. \end{aligned}$$

(7.8)

(4):

The following representation in terms of the constant connection coefficients $c_{jk}^{(\nu )}$ of Proposition 5.1 holds on $\mathbb {D}(u^c)$:

$$\begin{aligned} (\mathbb {S}_\nu )_{jk}= & {} \left\{ \begin{array}{ll} e^{2\pi i \lambda ^\prime _k}\alpha _k ~c_{jk}^{(\nu )},&{}\quad \quad j\prec k, u_j^c\ne u_k^c, \\ 1 &{} \quad j =k, \\ 0 &{} \quad j\succ k, u_j^c\ne u_k^c, \\ 0 &{}\quad j\ne k , u_j^c= u_k^c, \end{array} \right. ;\nonumber \\ (\mathbb {S}_{\nu +\mu }^{-1})_{jk}= & {} \left\{ \begin{array}{ll} 0 &{} \quad j\ne k, u_j^c= u_k^c, \\ 0 &{} \quad j\prec k, u_j^c\ne u_k^c, \\ 1 &{} \quad j =k, \\ -e^{2\pi i (\lambda ^\prime _k-\lambda ^\prime _j)}\alpha _k~c_{jk}^{(\nu )} &{} \quad j\succ k, u_j^c\ne u_k^c, \end{array} \right. \end{aligned}$$

(7.9)

where the relation $j\prec k$ is defined for $j\ne k$ such that $u_j^c\ne u_k^c$ and means that $\mathfrak {R}(z(u_j^c-u_k^c))<0$ when $\arg z=\tau$.

Remark 7.1

The above (7.9) generalizes Theorem 2.3 in the presence of isomonodromic deformation parameters, including coalescences. Notice that the ordering relation $\prec$ here is referred to $u^c$, while in Theorem 2.3 it refers to $u^0$.

Proof

We use the labelling (6.1)–(6.2) for $u^c$.

a) $\underline{\hbox {Case } \lambda ^\prime _k\not \in \mathbb {Z}}$.

• Construction of $\vec {Y}_k(z,u~|\nu )$. We have $\vec {\Psi }_k^{(sing)}(\lambda ,u|~\nu )=\vec {\Psi }_k(\lambda ,u|~\nu )$ and (7.4) is

  $$\begin{aligned} \vec {Y}_k(z,u~|\nu ):= \dfrac{1}{2\pi i } \int _{\gamma _k(\eta )} e^{z\lambda } \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda \end{aligned}$$

  (7.10)

Since $\vec {\Psi }_k(\lambda ,u~|\nu )$ grows at infinity no faster than some power of $\lambda$, the integral converges in a sector of amplitude at most $\pi$. Now, $\vec {\Psi }_k(\lambda ,u~|\nu )$ satisfies Theorem 5.1, hence if u varies in $\mathbb {D}(u^c)$ the following facts hold.

• (1) $\vec {\Psi }_k(\lambda ,u~|\nu )$ is branched at $\lambda =u_k$ and possibly at other poles $u_l$ such that $u_l^c\ne u_k^c$.

• (2) $\vec {\Psi }_k(\lambda ,u~|\nu )$ is holomorphic at all $\lambda = u_j$ such that $u_j^c = u_k^c$, $j\ne k$.

It follows from (1) to (2) that the path of integration can be modified: for $\alpha$ such that $u_k^c=\lambda _\alpha$, we have

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )= \dfrac{1}{2\pi i } \int _{\Gamma _\alpha (\eta )} e^{z\lambda } \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda , \end{aligned}$$

(7.11)

where $\Gamma _\alpha (\eta )$ is the path which comes from $\infty$ in direction $\eta -\pi$, encircles $\lambda _\alpha$ along $\partial \mathbb {D}_\alpha$ anti-clockwise and goes to $\infty$ in direction $\eta$. This path encloses all the $u_j$ such that $u_j^c=\lambda _\alpha$, end excludes the others. See Fig. 4. We conclude that u can vary in $\mathbb {D}(u^c)$ and the integral (7.11) converges for z in the sector

$$\begin{aligned} \mathcal {S}(\eta ):=\Bigl \{ z\in \mathcal {R}(\mathbb {C}\backslash \{0\})~\hbox { such that } \frac{\pi }{2}-\eta<\arg z <\frac{3\pi }{2}-\eta \Bigr \}, \end{aligned}$$

defining $\vec {Y}_k(z,u~|\nu )$ as a holomorphic function of $(z,u)\in \mathcal {S}(\eta )\times \mathbb {D}(u^c)$. Now, if u varies in $\mathbb {D}(u^c)$ and $\epsilon _0$ satisfies (5.1) none of the vectors $u_i-u_j$ such that $u_i^c=\lambda _\alpha$ and $u_j^c=\lambda _\beta$, $1\le \alpha \ne \beta \le s$, cross a direction $\eta$ mod $\pi$, for every $\eta _{\nu +1}<\eta <\eta _\nu$. Due to 1. and 2. above, a vector function $\vec {\Psi }_k(\lambda ,u~|\nu )$ is well defined in $\mathcal {P}_\eta$ and $\mathcal {P}_{\tilde{\eta }}$ for any $\eta _{\nu +1}<\eta<\tilde{\eta }< \eta _\nu$, and so on $\mathcal {P}_\eta \cup \mathcal {P}_{\tilde{\eta }}$. Therefore, the integral in (7.11) satisfies

$$\begin{aligned} \dfrac{1}{2\pi i } \int _{\Gamma _\alpha (\eta )}e^{z\lambda } \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda = \dfrac{1}{2\pi i } \int _{\Gamma _\alpha (\tilde{\eta })}e^{z\lambda } \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda , \quad \quad z\in \mathcal {S}(\eta )\cap \mathcal {S}(\tilde{\eta }), \end{aligned}$$

namely one is the analytic continuation of the other, so defining the function $\vec {Y}_k(z,u~|\nu )$ as analytic on $\widehat{\mathcal {S}}_\nu \times \mathbb {D}(u^c)$, where

$$\begin{aligned} \widehat{\mathcal {S}}_\nu :=\bigcup _{\eta _{\nu +1}<\eta <\eta _\nu }\mathcal {S}(\eta ) \end{aligned}$$

coincides with (7.3) (with $h=0$).

Fig. 4

The paths of integration $\Gamma _\alpha$, $\Gamma _\beta$, etc $\alpha ,\beta , \ldots \in \{1,\ldots ,s\}$

Finally, notice that $e^{\lambda z}(\lambda -\Lambda ) \vec {\Psi }_k(\lambda ,u~|\nu )\Bigl |_{\Gamma (\alpha )}=0$, due to the exponential factor. By (2.25), the vector solutions $\vec {Y}_k(z,u~|\nu )$ satisfy system (1.1).

• Asymptotic behaviour. From (5.4)–(5.5), we write (7.11) as

  $$\begin{aligned} \vec {Y}_k(z,u~|\nu )= \dfrac{1}{2\pi i } \int _{\Gamma _\alpha (\eta )} e^{z\lambda } \Bigl (\Gamma (\lambda ^\prime _j+1)\vec {e}_j+\sum _{l\ge 1} \vec {b}_l^{(k)}(u)(\lambda -u_k)^l\Bigr )(\lambda -u_k)^{-\lambda ^\prime _k-1}, \mathrm{d}\lambda . \end{aligned}$$

with holomorphic $\vec {b}_l^{(k)}(u)$ on $\mathbb {D}(u^c)$. We split the series as $\sum _{l\ge 1}=\sum _{l=1}^{\mathcal {N}}+\sum _{l\ge \mathcal {N}+1}$, and recall the standard formula (see [18])

$$\begin{aligned} \int _{\Gamma _\alpha (\eta )}(\lambda -\lambda _k)^a e^{z\lambda } \mathrm{d}\lambda = \int _{\gamma _k(\eta )}(\lambda -\lambda _k)^a e^{z\lambda } \mathrm{d}\lambda =\frac{z^{-a-1} e^{\lambda _k z}}{\Gamma (-a)} \end{aligned}$$

so that

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )=\left( \vec {e}_k+\sum _{l=1}^{\mathcal {N}} \frac{\vec {b}_l^{(k)}(u)}{\Gamma (\lambda ^\prime _k+1-l)}z^{-l}+R_{\mathcal {N}}(z)\right) ~z^{\lambda ^\prime _k}e^{\lambda _k z}, \end{aligned}$$

with remainder

$$\begin{aligned} R_{\mathcal {N}}(z) = \oint _{\Gamma _0(\eta )}\sum _{l\ge \mathcal {N}} \frac{\vec {b}_l^{(k)}(u) }{z^l} ~e^x x^{l-\lambda ^\prime _k-1} ~\mathrm{d}x~=O(z^{-\mathcal {N}+1}). \end{aligned}$$

The integral is along a path $\Gamma _0(\eta )$, coming from $\infty$ along the left part of the half line oriented from 0 to $\infty$ in direction $\eta +\arg z$, going around 0, and back to $\infty$ along the right part. The estimate $O(z^{-\mathcal {N}+1})$ is standard. We conclude that

$$\begin{aligned}&\vec {Y}_k(z,u~|\nu ) \left( z^{\lambda ^\prime _k}e^{u_k z}\right) ^{-1} \sim \vec {e}_k+\sum _{l=1}^{\infty } \frac{\vec {b}_l^{(k)}(u)}{\Gamma (\lambda ^\prime _k+1-l)}z^{-l}\equiv \vec {e}_k+\sum _{l=1}^{\infty } \vec {f}_l^{~(k)}(u) z^{-l},\\&\quad z\rightarrow \infty \hbox { in } \widehat{\mathcal {S}}_\nu \end{aligned}$$

with

$$\begin{aligned} \vec {f}_l^{~(k)}(u):= \frac{\vec {b}_l^{(k)}(u)}{\Gamma (\lambda ^\prime _k+1-l)}. \end{aligned}$$

(7.12)

b) $\underline{\hbox {Case } \lambda ^\prime _k\in \mathbb {N}=\{0,1,2,\ldots \}}$.

• Construction of $\vec {Y}_k(z,u~|\nu )$. Definition (7.4) is

  $$\begin{aligned} \vec {Y}_k(z,u~|\nu )&:= \frac{1}{2\pi i} \int _{\gamma _k(\eta )} e^{z\lambda } \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu ) \mathrm{d}\lambda \\&\underset{(5.10)}{=} \frac{1}{2\pi i} \int _{\gamma _k(\eta )} e^{z\lambda } \left( \frac{\vec {\psi }_k(\lambda ,u~|\nu )}{(\lambda -u_k)^{\lambda ^\prime _k+1}}+ \vec {\Psi }_k(\lambda ,u~|\nu )\ln (\lambda -u_k) \right) \mathrm{d}\lambda . \end{aligned}$$

The same facts 1. and 2. of the previous case apply to $\vec {\Psi }_k(\lambda ,u~|\nu )$ and $\vec {\psi }_k(\lambda ,u~|\nu )$ and allow to rewrite

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )&= \frac{1}{2\pi i} \int _{\Gamma _\alpha (\eta )} e^{z\lambda } \left( \frac{\vec {\psi }_k(\lambda ,u~|\nu )}{(\lambda -u_k)^{\lambda ^\prime _k+1}}+ \vec {\Psi }_k(\lambda ,u~|\nu )\ln (\lambda -u_k) \right) \mathrm{d}\lambda \\&= \frac{1}{2\pi i} \int _{\Gamma _\alpha (\eta )} e^{z\lambda } \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu ) \mathrm{d}\lambda . \end{aligned}$$

We conclude that $\vec {Y}_k(z,u~|\nu )$ is analytic on $\widehat{\mathcal {S}}_\nu \times \mathbb {D}(u^c)$. Moreover, $e^{\lambda z}(\lambda -\Lambda ) \vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu )\Bigl |_{\Gamma (\alpha )}=0$, due to the exponential factor. By (2.25), the vector solution $\vec {Y}_k(z,u~|\nu )$ satisfies the system (1.1).

• Asymptotic behaviour. By (5.7) and (5.11), and the fact that $\vec {\psi }_k$ has no singularities at $u_j\in \mathbb {D}_\alpha$, $j\ne k$, so that the terms $\sum _{l\ge 1+\lambda ^\prime _k} \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l$ in $\vec {\psi }_k(\lambda ,u~|\nu )$ do not contribute to the integration, we can write

  $$\begin{aligned}&\vec {Y}_k(z,u~|\nu )\\&\quad = \frac{1}{2\pi i} \int _{\Gamma _\alpha (\eta )} \left( \frac{ \lambda ^\prime _k!\vec {e}_k+ \sum _{l=1}^{\lambda ^\prime _k} \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l}{(\lambda -u_k)^{\lambda ^\prime _k+1}}+ \sum _{l=0}^\infty \vec {d}_l^{~(k)}(u)(\lambda -u_k)^l~\ln (\lambda -u_k) \right) e^{z\lambda }~\mathrm{d}\lambda . \end{aligned}$$

By Cauchy formula

$$\begin{aligned}&\frac{1}{2\pi i} \int _{\Gamma _\alpha (\eta )} \left( \frac{ \lambda ^\prime _k!\vec {e}_k+ \sum _{l=1}^{\lambda ^\prime _k} \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l}{(\lambda -u_k)^{\lambda ^\prime _k+1}}\right) e^{z\lambda }~\mathrm{d}\lambda \\&\quad = \frac{1}{\lambda ^\prime _k!} \frac{\mathrm{d}^{\lambda ^\prime _k}}{\mathrm{d}\lambda ^{\lambda ^\prime _k}}\left. \left[ \left( \lambda ^\prime _k!\vec {e}_k+ \sum _{l=1}^{\lambda ^\prime _k} \vec {b}_l^{~(k)}(u) (\lambda -u_k)^l\right) e^{z\lambda } \right] \right| _{\lambda =u_k} \\&\quad = z^{\lambda ^\prime _k} e^{u_k z}\left( \vec {e}_k+ \sum _{l=1}^{\lambda ^\prime _k} \vec {f}_l^{~(k)}(u) \frac{1}{z^l} \right) , \end{aligned}$$

where

$$\begin{aligned} \vec {f}_l^{~(k)}(u):=\frac{\vec {b}_l^{(k)}(u)}{(\lambda ^\prime _k-l)!},\quad l=1,\ldots ,\lambda ^\prime _k. \end{aligned}$$

(7.13)

In order to evaluate the terms with logarithm, we observe that for any function $g(\lambda )$ holomorphic along $L_k(\eta )$, including $\lambda =u_k$, we have

$$\begin{aligned} \int _{\gamma _k(\eta )}g(\lambda )\ln (\lambda -u_k)\mathrm{d}\lambda = \int _{L_k(\eta )^{-}} g(\lambda )\ln (\lambda -u_k)_{-}\mathrm{d}\lambda - \int _{L_k(\eta )^{+}} g(\lambda )\ln (\lambda -u_k)_{+}\mathrm{d}\lambda , \end{aligned}$$

where $L_k(\eta )^{+}$ and $L_k(\eta )^{-}$, respectively, are the left and right parts of $L_k(\eta )$, oriented from 0 to $\infty$. Since $\ln (\lambda -u_k)_{+}=\ln (\lambda -u_k)_{+}-2\pi i$, we conclude that

$$\begin{aligned} \int _{\gamma _k(\eta )}g(\lambda )\ln (\lambda -u_k)\mathrm{d}\lambda = 2\pi i \int _{L_k(\eta )} g(\lambda )\mathrm{d}\lambda . \end{aligned}$$

(7.14)

Keeping into account that the integral along $\Gamma _\alpha$ can be interchanged with that along $\gamma _k$, it follows that

$$\begin{aligned} \frac{1}{2\pi i} \int _{\Gamma _\alpha (\eta )} \vec {\Psi }_k(\lambda ,u~|\nu )\ln (\lambda -u_k) e^{z\lambda }~\mathrm{d}\lambda&= \int _{L_k(\eta )} \vec {\Psi }_k(\lambda ,u~|\nu ) e^{z\lambda }~\mathrm{d}\lambda \\&=\int _{L_k(\eta )} \sum _{l=0}^\infty \vec {d}_l^{~(k)}(u)(\lambda -u_k)^l~ e^{z\lambda }~\mathrm{d}\lambda . \end{aligned}$$

We conclude, by the standard evaluation of the remainder analogous to $R_{\mathcal {N}}(z)$ considered before, and the variation of $\eta$ in the range $(\eta _{\nu +1},\eta _\nu )$, that²⁰

$$\begin{aligned} \int _{L_k(\eta )} \vec {\Psi }_k(\lambda ,u~|\nu ) e^{z\lambda }~\mathrm{d}\lambda&\sim e^{u_k z}\left( \sum _{l=0}^\infty (-1)^{l+1} l! ~ \vec {d}_l^{(k)}(u) ~z^{-l-1}\right) ,\quad \quad z\rightarrow \infty \hbox { in } \widehat{\mathcal {S}}_\nu . \\&=z^{\lambda ^\prime _k}e^{u_k z}\left( \sum _{l=\lambda ^\prime _k+1}^\infty \vec {f}_l^{~(k)}(u)~z^{-l}\right) , \end{aligned}$$

where

$$\begin{aligned} \vec {f}_l^{~(k)}(u):=(-1)^{l-\lambda ^\prime _k}(l-\lambda ^\prime _k-1)!~\vec {d}_{l-\lambda ^\prime _k-1}^{~(k)}(u) ,\quad \quad l\ge \lambda ^\prime _k+1. \end{aligned}$$

(7.15)

In conclusion, we have the expansion

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )\sim z^{\lambda ^\prime _k}e^{u_k z}\left( \vec {e}_k+\sum _{l=1}^\infty \vec {f}_l^{~(k)}(u)~z^{-l}\right) , \quad \quad z\rightarrow \infty \hbox { in } \widehat{\mathcal {S}}_\nu , \end{aligned}$$

with coefficients $\vec {f}_l^{~(k)}(u)$ holomorphic in $\mathbb {D}(u^c)$ defined in (7.13)–(7.15). Notice that, in exceptional cases, $\vec {\Psi }_k$ may be identically zero, so that

$$\begin{aligned} \vec {f}_l^{~(k)}=0\hbox { for } l\ge \lambda ^\prime _k+1. \end{aligned}$$

(7.16)

c) $\underline{\hbox {Case } \lambda ^\prime _k\in \mathbb {Z}_{-}=\{-1,-2,\ldots \}}$

• Construction of $\vec {Y}_k(z,u~|\nu )$. Definition (7.5) is

  $$\begin{aligned} \vec {Y}_k(z,u~|\nu ):= \int _{L_k(\eta )} e^{\lambda z} \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda \equiv \int _{\mathcal {L}_\alpha (\eta )} e^{\lambda z} \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda . \end{aligned}$$

In the last equality, we have used the fact that $\vec {\Psi }_k(\lambda ,u~|\nu )$ is analytic in $\mathbb {D}_\alpha \times \mathbb {D}(u^c)$, where $\lambda _\alpha =u_k^c$.

We conclude analogously to previous cases that $\vec {Y}_k(z,u~|\nu )$ is analytic in $\widehat{\mathcal {S}}_\nu \times \mathbb {D}(u^c)$. It is a solution of (1.1), by (2.25), because $\vec {\Psi }_k(\lambda ,u~|\nu )$ is analytic at $\lambda =u_k$ and behaves as in (5.4)–(5.5), so that

$$\begin{aligned} e^{\lambda z} (\lambda I - \Lambda (u)) \vec {\Psi }_k(\lambda ,u~|\nu )\Bigr |_{\mathcal {L}_\alpha }= & {} e^{\lambda z} (\lambda I - \Lambda (u)) \vec {\Psi }_k(\lambda ,u~|\nu )\Bigr |_{L_k}\\= & {} 0-(u_kI-\Lambda (u)) \vec {\Psi }_k(\lambda ,u_k~|\nu )=0. \end{aligned}$$

• Asymptotic behaviour. We have, from (5.4)–(5.5),

  $$\begin{aligned} \vec {Y}_k(z,u~|\nu )= \int _{\mathcal {L}_\alpha (\eta )} e^{\lambda z} \left( \frac{(-1)^{\lambda ^\prime _k} ~\vec {e}_k}{(-\lambda ^\prime _k-1)!} (\lambda -u_k)^{-\lambda ^\prime _k-1}+\sum _{l\ge 1} \vec {b}_l^{(k)}(u)(\lambda -u_k)^{l-\lambda ^\prime _k-1} \right) \mathrm{d}\lambda \end{aligned}$$

We integrate term by term in order to obtain the asymptotic expansion (the remainder for the truncated series is evaluate in standard way, as $R_{\mathcal {N}}(z)$ above). For the integration, we use

$$\begin{aligned}&\int _{L_k(\eta )} (\lambda -u_k)^m e^{\lambda z} \mathrm{d}\lambda =\frac{e^{u_kz}}{z^{m+1}}\int _{+\infty e^{i\phi }}^0 x^m e^x \mathrm{d}x = \frac{e^{u_kz}}{z^{m+1}} m!~(-1)^{m+1}, \\&\quad \frac{\pi }{2}<\phi <\frac{3\pi }{2}. \end{aligned}$$

We obtain, analogously to previous cases,

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )\sim z^{\lambda ^\prime _k}e^{u_k z}\left( \vec {e}_k + \sum _{l=1}^\infty \vec {f}_l^{~(k)}(u) z^{-l}\right) ,\quad \quad z\rightarrow \infty \hbox { in } \widehat{\mathcal {S}}_\nu , \end{aligned}$$

where the holomorphic in $\mathbb {D}(u^c)$ coefficients are

$$\begin{aligned} \vec {f}_l^{~(k)}(u):= (-1)^{l-\lambda ^\prime _k} (l-\lambda ^\prime _k-1)!~\vec {b}_l^{(k)}(u). \end{aligned}$$

(7.17)

Remark 7.2

We cannot use $\vec {\Psi }_k^{(\mathrm{sing})}(\lambda ,u~|\nu )$ in (5.8) to define $\vec {Y}_k(z,u~|\nu )$ if u varies in the whole $\mathbb {D}(u^c)$. On the other hand, if u is restricted to a $\tau$-cell, so that the eigenvalues $u_j$ are all distinct, by (7.14) we can write

$$\begin{aligned} \vec {Y}_k(z,u~|\nu )= \int _{L_k(\eta )} e^{\lambda z} \vec {\Psi }_k(\lambda ,u~|\nu ) \mathrm{d}\lambda \underset{(7.14)}{=} \frac{1}{2\pi i } \int _{\gamma _k(u)} e^{\lambda z}\vec {\Psi }_k(\lambda ,u~|\nu ) \ln (\lambda -u_k)\mathrm{d}\lambda . \end{aligned}$$

Then, we can use the local expansion (5.9) and the fact that $\int _{\gamma _k(u)} \hbox {reg}(\lambda -u_k)\mathrm{d}\lambda =0$, receiving

$$\begin{aligned} \vec {Y}_k(z,u~|\nu ) =\frac{1}{2\pi i } \int _{\gamma _k(u)} e^{\lambda z}\vec {\Psi }_k^{(sing)}(\lambda ,u~|\nu ) \mathrm{d}\lambda \end{aligned}$$

7.1 Fundamental matrix solutions

The vector solutions $\vec {Y}_k(z,u~|\nu )$ constructed above can be arranged as columns of the matrix

$$\begin{aligned} Y_\nu (z,u):=\Bigl [\vec {Y}_k(z,u~|\nu )~\Bigl |~\cdots ~\Bigr |~\vec {Y}_n(z,u~|\nu )\Bigr ], \end{aligned}$$

which thus solves system (1.1). From the general theory of differential systems, it admits analytic continuation as analytic matrix valued function on $\mathcal {R}(\mathbb {C}\backslash \{0\})\times \mathbb {D}(u^c)$. Letting $B=\hbox {diag} A=\hbox {diag}(\lambda ^\prime _1,\ldots ,\lambda ^\prime _n)$, the asymptotic expansions obtained above are summarized as

$$\begin{aligned} Y_\nu (z,u~|\nu ) ~z^{-B}e^{-\Lambda (u) z}\sim F(z,u)= & {} I+\sum _{l=1}^\infty F_l(u) z^{-l},\quad \quad z\rightarrow \infty \hbox { in } \widehat{\mathcal {S}}_\nu ,\\ F_l(u)= & {} \Bigl [\vec {f}_l^{~(1)}(u)~|~\cdots ~|~\vec {f}_l^{~(n)}(u)\Bigr ]. \end{aligned}$$

Therefore, the coefficients $F_l(u)$ of the formal solution $Y_F(z,u)=F(z,u)z^{B}e^{\Lambda (u) z}$ are holomorphic in $\mathbb {D}(u^c)$. Moreover, the leading term is the identity I, which implies that $Y_\nu (z,u)$ is a fundamental matrix solution.

Consider now another direction $\eta$, satisfying $\eta _{\nu +\mu +1}<\eta <\eta _{\nu +\mu }$. The above discussion can be repeated. We obtain a fundamental matrix solution $Y_{\nu +\mu }(z,u)$ with canonical asymptotics $Y_F(z,u)$ in $\widehat{S}_{\nu +\mu }$. Again, for $\eta$ satisfying $\eta _{\nu +2\mu +1}<\eta <\eta _{\nu +2\mu }$ we obtain the analogous result for $Y_{\nu +2\mu }(z,u)$ with canonical asymptotics in $\widehat{S}_{\nu +2\mu }$. This can be repeated for every $\nu +h\mu$, $h\in \mathbb {Z}$, obtaining the fundamental matrix solutions $Y_{\nu +h\mu }(z,u)$ with canonical asymptotics $Y_F(z,u)$ in $\widehat{S}_{\nu +h\mu }$. So, Points (1) and (2) of Theorem 7.1 are proved.

Stokes matrices are defined by (7.7). Thus, $\mathbb {S}_{\nu +h\mu }(u)=Y_{\nu +h\mu }(z,u)^{-1}Y_{\nu +(h+1)\mu }(z,u)$ is holomorphic in $\mathbb {D}(u^c)$. Let us consider the relations for $h=0,1$:

$$\begin{aligned} Y_{\nu +\mu }(z,u)=Y_\nu (z,u) \mathbb {S}_\nu (u), \quad \quad Y_{\nu +2\mu }(z,u)=Y_{\nu +\mu }(z,u) \mathbb {S}_{\nu +\mu }(u). \end{aligned}$$

(7.18)

Let u be fixed in a $\tau$-cell, so that $\Lambda$ has distinct eigenvalues. From Theorem 2.3 at the fixed u we receive

$$\begin{aligned} \bigl ( \mathbb {S}_\nu (u)\bigr )_{jk}= & {} \left\{ \begin{array}{cc} e^{2\pi i \lambda ^\prime _k}\alpha _k ~c_{jk}^{(\nu )}&{} ~~~\hbox { for } j\prec k, \\ \\ 1 &{} ~~~\hbox { for } j =k, \\ \\ 0 &{} ~~~\hbox { for } j\succ k, \end{array} \right. \\ \bigl ( \mathbb {S}_{\nu +\mu }^{-1}(u)\bigr )_{jk}= & {} \left\{ \begin{array}{cc} 0 &{} ~~~\hbox { for } j\prec k, \\ \\ 1 &{}~~~ \hbox { for } j =k, \\ \\ -e^{2\pi i (\lambda ^\prime _k-\lambda ^\prime _j)}\alpha _k~c_{jk}^{(\nu )} &{} ~~~\hbox { for } j\succ k. \end{array} \right. \end{aligned}$$

Here, for $j\ne k$ the ordering relation $j\prec k$ $\Longleftrightarrow$ $\mathfrak {R}(z(u_j-u_k))|_{\arg z=\tau }<0$ is well defined for every u in the $\tau$-cell, because no Stokes rays $\mathfrak {R}(z(u_j-u_k))=0$ cross $\arg z=\tau$ as u varies in the $\tau$-cell.

The relation $j\prec k$ may change to $j\succ k$ when passing from one $\tau$-cell to another only for a pair $u_j$, $u_k$ such that $u_j^c=u_k^c$. This is due to the choice of $\epsilon _0$ as in (5.1). On the other hand, $c_{jk}^{(\nu )}=0$ whenever $u_j^c=u_k^c$. This means that (7.9) is true at every fixed u in every $\tau$-cell, with ordering relation $j\prec k$ defined for $j\ne k$ such that $u_j^c\ne u_k^c$, namely $\mathfrak {R}(z(u_j^c-u_k^c))<0$ when $\arg z=\tau$.

Since the $\mathbb {S}_{\nu +h\mu }$ are holomorphic in $\mathbb {D}(u^c)$ and the $c_{jk}^{(\nu )}$ are constant in $\mathbb {D}(u^c)$, we conclude that Stokes matrices are constant in $\mathbb {D}(u^c)$ and hence (7.9) holds in $\mathbb {D}(u^c)$. The vanishing conditions (7.8) follow from the vanishing conditions (5.14) for the connection coefficients, plus the fact that we can generate all the $\mathbb {S}_{\nu +h\mu }$ from the formula $\mathbb {S}_{\nu +2\mu }=e^{-2\pi i B}\mathbb {S}_\nu e^{2\pi i B}$. $\square$

8 Non-uniqueness at $u=u^c$ of the formal solution

By Laplace transform, we prove Corollary 2.1 in Background 1, asserting that system (2.19) has unique formal solution if and only if the constant diagonal entries of A(u) satisfy the partial non-resonance

$$\begin{aligned} \lambda ^\prime _i-\lambda ^\prime _j\not \in \mathbb {Z}\backslash \{0\} \quad \hbox { for every } i\ne j\hbox { such that } u_i^c=u_j^c. \end{aligned}$$

Otherwise, the Laplace transform will be proved to generate a family of formal solutions

$$\begin{aligned} \mathring{Y}_F(z)=\Bigl (I+\sum _{l=1}^\infty \mathring{F}_l z^{-l}\Bigr )z^Be^{\Lambda (u^c) z}, \end{aligned}$$

whose coefficients $\mathring{F}_l$ depend on a finite number of arbitrary parameters.

Due to the strategy of Sect. 6.6, it will suffice to consider the generic case when all $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n ~\not \in \mathbb {Z}$ and A has no integer eigenvalues. Indeed, if this is not the case, the gauge transformation (6.46) relates a formal solution ${}_\gamma Y_F$ to $Y_F$ at any point u, through (6.48), so that the coefficients $F_l$ of a formal expansion do not depend on $\gamma$. We are interested in these coefficients.

Consider system (1.4) under the assumptions that it is (strongly) isomonodromic in $\mathbb {D}(u^c)$, so that $(A)_{ij}(u^c)=0$ for $u_i^c=u_j^c$. For simplicity, we order the eigenvalues as in (6.1)–(6.2). Since $B_1(u)$, ..., $B_n(u)$ are holomorphic at $u^c$, system (1.4) at $u=u^c$ is

$$\begin{aligned} \frac{\mathrm{d}\Psi }{\mathrm{d}\lambda } = \left( \frac{\sum _{j=1}^{p_1}B_j(u^c)}{\lambda -\lambda _1}+\frac{\sum _{j=p_1+1}^{p_1+p_2}B_j(u^c)}{\lambda -\lambda _2} + \cdots + \frac{\sum _{j=p_1+\cdots +p_{s-1}+1}^n B_j(u^c)}{\lambda -\lambda _s} \right) \Psi \end{aligned}$$

(8.1)

Let $G^{(\varvec{p}_1)}$ be as in (6.24). The gauge transformation $\Psi (\lambda )=G^{(\varvec{p}_1)}\widetilde{\Psi }(\lambda )$ yields

$$\begin{aligned} \frac{\mathrm{d}\widetilde{\Psi }}{\mathrm{d}\lambda }=\left( \frac{T^{(\varvec{p}_1)}}{\lambda -\lambda _1}+\sum _{\alpha =2}^s \frac{D_\alpha ^{(\varvec{p}_1)}}{\lambda -\lambda _\alpha } \right) \widetilde{\Psi }, \end{aligned}$$

(8.2)

where

$$\begin{aligned} T^{(\varvec{p}_1)}:=T^{(1)}+\cdots +T^{(p_1)}=\hbox {diag}\left( -\lambda ^\prime _1-1,\ldots ,-\lambda ^\prime _{p_1}-1,~\underbrace{0,\ldots ,~0}_{n-p_1}\right) . \end{aligned}$$

and $D_\alpha ^{(\varvec{p}_1)}:= {G^{(\varvec{p}_1)}}^{-1} \cdot \sum _{j=p_1+\cdots +p_{\alpha -1}+1}^{p_1+\cdots +p_{\alpha }} B_j(u^c) \cdot G^{(\varvec{p}_1)}$. The matrix coefficient in system (8.2) has convergent Taylor series at $\lambda =\lambda _1$

$$\begin{aligned} \frac{\mathrm{d}\widetilde{\Psi }}{\mathrm{d}\lambda }=\frac{1}{\lambda -\lambda _1}\left( T^{(\varvec{p}_1)}+ \sum _{m=1}^\infty \mathfrak {D}_m(\lambda -\lambda _1)^m \right) \widetilde{\Psi }, \quad \quad \mathfrak {D}_m= \sum _{\alpha =2}^s \frac{(-1)^{m+1}}{(\lambda _1-\lambda _\alpha )^m}D_\alpha ^{(\varvec{p}_1)}. \end{aligned}$$

We consider $\eta _{\nu +1}<\eta <\eta _\nu$ and $\lambda$ in the plane with branch cuts $\mathcal {L}_\alpha = \mathcal {L}_\alpha (\eta )$ issuing from $\lambda _1,\ldots ,\lambda _s$ to infinity in direction $\eta$, as in (5.2). Close to the Fuchsian singularity $\lambda =\lambda _1$ a fundamental matrix solution to (8.1) has Levelt form

$$\begin{aligned} \mathring{\Psi }^{(\varvec{p}_1)}(\lambda )=G^{(\varvec{p}_1)}\Bigl ( I+\sum _{l=1}^\infty \mathfrak {G}_l (\lambda -\lambda _1)^l \Bigr )(\lambda -\lambda _1)^{T^{(\varvec{p}_1)}}, \end{aligned}$$

(8.3)

where the matrix entries $( \mathfrak {G}_l)_{ij}$, $1\le i\le j\le n$, are recursively computed by the following formulae (see Appendix C for an explanation of (8.3), or [27, 62]).

• If $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}= l$ positive integer, $( \mathfrak {G}_l)_{ij}$ is arbitrary.

• If $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}\ne l$ (positive integer)

  $$\begin{aligned} ( \mathfrak {G}_l)_{ij} =\frac{1}{T^{(\varvec{p}_1)}_{jj}-T^{(\varvec{p}_1)}_{ii}+l}\left( \sum _{p=1}^{l-1} \mathfrak {D}_{l-p}\mathfrak {G}_l+\mathfrak {D}_l \right) _{ij} \quad \hbox {(sum is zero for } l=1). \end{aligned}$$

Since we have assumed that all the $\lambda ^\prime _k$ are not integers, the only possibility to have $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}= l$ occurs for $1\le i,j\le p_1$, precisely

$$\begin{aligned} T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}=\lambda ^\prime _j-\lambda ^\prime _i=l. \end{aligned}$$

(8.4)

In this case, (8.3) is a family depending on a finite number of parameters due to the arbitrary $(\mathfrak {G}_l)_{ij}$. Thus, in the first $p_1$ columns of a solution of type (8.3)

$$\begin{aligned} \vec {\mathring{\Psi }}_j(\lambda ~|\nu )=\Bigl (\Gamma (\lambda ^\prime _k+1)\vec {e}_k +\sum _{l=1}^\infty \mathring{b}_l^{(j)} (\lambda -\lambda _1)\Bigr )(\lambda -\lambda _1)^{-\lambda ^\prime _j-1},\quad \quad j=1,\ldots ,p_1. \end{aligned}$$

the vectors $\mathring{b}_l^{(j)}$ contain a finite number of parameters. The Laplace transform

$$\begin{aligned} \vec {\mathring{Y}}_j (z~|\nu )=\int _{\Gamma _1(\eta )} e^{z\lambda } \vec {\mathring{\Psi }}_j(\lambda ~|\nu ) \mathrm{d}\lambda ,\quad \quad j=1,\ldots ,p_1, \end{aligned}$$

yields the first $p_1$ columns of a fundamental matrix solution of (2.19). Repeating the same computations of Section 7, we obtain, for $j=1,\ldots ,p_1$,

$$\begin{aligned} \vec {\mathring{Y}}_j (z~|\nu ) ~z^{-\lambda ^\prime _j}e^{-\lambda _1 z}\sim \vec {e}_j +\sum _{l=1}^\infty \frac{ \mathring{b}_l^{(j)} }{\Gamma (\lambda ^\prime _j+1-l)} \frac{1}{z^l}, \quad z\rightarrow \infty \hbox { in } \mathcal {S}_\nu , \end{aligned}$$

where $\mathcal {S}_\nu$ is given in (7.1). We repeat the same construction at all $\lambda _1$, ..., $\lambda _s$. This yields a family of fundamental matrix solutions of (2.19)

$$\begin{aligned} \mathring{Y}_\nu (z) = \Bigl [ \vec {\mathring{Y}}_1 (z~|\nu )~|~\cdots ~|~ \vec {\mathring{Y}}_n (z~|\nu ) \Bigr ], \end{aligned}$$

depending on a finite number of parameters, with the behaviour for $z\rightarrow \infty$ in ${S}_\nu$

$$\begin{aligned} \mathring{Y}_\nu (z)\sim \mathring{Y}_F(z)= & {} \Bigl (I+\sum _{l=1}^\infty \mathring{F}_lz^{-l} \Bigr ) z^B e^{\Lambda (u^c) z}; \quad \quad \mathring{F}_l= \Bigl [ \vec {\mathring{f}}_1^{~(l)} ~|~\cdots ~|~ \vec {\mathring{f}}_n^{~(l)} \Bigr ],\\ \vec {\mathring{f}}_j^{~(l)}= & {} \frac{\vec {\mathring{b}}_j^{~(l)}}{\Gamma (\lambda ^\prime _j+1-l)}. \end{aligned}$$

We conclude that the formal solution is not unique whenever a condition (8.4) occurs. Only one element in the family satisfies $\mathring{Y}_F(z)=Y_F(z,u^c)$.

Remark 8.1

If we choose one formal solution $\mathring{Y}_F(z)$, then the corresponding $\mathring{Y}_\nu (z)$ with asymptotic expansion $\mathring{Y}_F(z)$ in $\mathcal {S}_\nu$ is unique. For more details on the Stokes phenomenon at $u=u^c$, see [13].

Appendices

Appendix A. non-normalized Schlesinger system

Lemma 9.1

The integrability condition $dP=P\wedge P$ of the Pfaffian system (3.2) defined on a polydisc $\mathbb {D}(u^0)$ contained in a $\tau$-cell is the non-normalized Schlesinger system (3.3)–(3.5).

Proof

For every $i\in \{1,\ldots ,n\}$, the Pfaffian system (3.2) can be rewritten as

$$\begin{aligned} P&= \left( \frac{B_i}{\lambda -u_i}+\sum _{j\ne i}\frac{B_j}{\lambda -u_j}\right) d(\lambda -u_i) +\sum _{j\ne i} \left( \gamma _j-\frac{B_j}{\lambda -u_j}\right) d(u_j-u_i)\\&\quad +\sum _{j=1}^n \gamma _j(u) \mathrm{d}\lambda . \end{aligned}$$

We study $\lambda -u_i\rightarrow 0$, while $u_j-u_i\ne 0$ for $j\ne i$ in $\mathbb {D}(u^0)$. In new variables

$$\begin{aligned} \lambda =\lambda , \quad \quad y_i=\lambda -u_i, \quad y_j=u_j-u_i,\quad j\ne i, \end{aligned}$$

P is rewritten in the following way (which defines the matrices $\mathcal {A}_j(y)$)

$$\begin{aligned} P=&\left( \frac{B_i}{y_i}+\sum _{j\ne i}\frac{B_j}{y_i-y_j}\right) \mathrm{d}y_i +\sum _{j\ne i} \left( \gamma _j-\frac{B_j}{y_i-y_j}\right) \mathrm{d}y_j+\sum _{j=1}^n \gamma _j(y) \mathrm{d}\lambda \\&=: \mathcal {A}_i(y) \mathrm{d}y_i +\sum _{j\ne i} \mathcal {A}_j(y) \mathrm{d}y_j +\sum _{j=1}^n \gamma _j(y) \mathrm{d}\lambda . \end{aligned}$$

The only singular term at $y_i=0$ is ${B_i/y_i}$ in $\mathcal {A}_i(y)$. The components relative to $\mathrm{d}y_1,\ldots ,\mathrm{d}y_n$ of $dP=P\wedge P$ are

$$\begin{aligned} \frac{\partial {\mathcal {A}}_l}{\partial y_k}+{\mathcal {A}}_l{\mathcal {A}}_k=\frac{\partial {\mathcal {A}}_k}{\partial y_l}+{\mathcal {A}}_k{\mathcal {A}}_l,\quad k\ne l, \end{aligned}$$

(9.1)

For $k\ne i$ and $l=i$, from (9.1), we receive

$$\begin{aligned} \frac{\partial }{\partial y_k} \left( \frac{B_i}{y_i}+\hbox {reg}(y_i)\right) + \left( \frac{B_i}{y_i}+\hbox {reg}(y_i)\right) {\mathcal {A}}_k= \frac{\partial {\mathcal {A}}_k}{\partial y_i} + {\mathcal {A}}_k\left( \frac{B_i}{y_i}+\hbox {reg}(y_i)\right) , \end{aligned}$$

where $\hbox {reg}(y_i)$ stands for an analytic term at $y_i=0$. We expand the above in Taylor series at $y_i=0$. The singular term (the residue at $y_i=0$) is

$$\begin{aligned} \frac{\partial B_i}{\partial y_k}=\bigl [\mathcal {A}_k|_{y_i=0},B_i\bigr ]= \frac{[B_k,B_i]}{u_k-u_i}+[\gamma _k,B_i],\quad k\ne i. \end{aligned}$$

(9.2)

The above gives the non-normalized Schlesinger Eqs. (3.4)–(3.5), because

$$\begin{aligned}&\frac{\partial B_i}{\partial y_k}=\frac{\partial B_i}{\partial (u_k-u_i)}= \frac{\partial u_k}{\partial (u_k-u_i)}\frac{\partial B_i }{\partial u_k}= \frac{\partial B_i}{\partial u_k}, \end{aligned}$$

(9.3)

$$\begin{aligned}&\frac{\partial B_i}{\partial u_i}= \sum _{k\ne i} \frac{\partial (u_k-u_i)}{\partial u_i} \frac{\partial B_i}{\partial (u_k-u_i)}= -\sum _{k\ne i} \frac{\partial B_i}{\partial u_k}\quad \Longrightarrow \quad \sum _{k=1}^n \frac{\partial B_i}{\partial u_k}=0. \end{aligned}$$

(9.4)

If we write the components of $dP=P\wedge P$ referring to $\mathrm{d}y_l$ ad $\mathrm{d}\lambda$, and we substitute into them (9.3)–(9.4), we receive (3.3), namely $\partial _l\gamma _k-\partial _k\gamma _l=\gamma _l\gamma _k-\gamma _k\gamma _l$. $\square$

Corollary 9.1

A solution $B_i(u)$, $i=1,\ldots ,n$, of (3.3)–(3.5) is holomorphically similar to a constant Jordan form on $\mathbb {D}(u^0)$. The similarity is realized by a fundamental matrix solution $G^{(i)}(u)$ of the Pfaffian system (9.6).

Proof

We must show that there exists a holomorphically invertible $G^{(i)}(u)$ on $\mathbb {D}(u^0)$ such that $(G^{(i)})^{-1} B_i G^{(i)}$ is a constant Jordan form. The conditions (9.1) for $k,l\ne i$ can be evaluated at $y_i=0$, and become

$$\begin{aligned} \frac{\partial {\mathcal {A}}_l |_{y_i=0}}{\partial y_k}+{\mathcal {A}}_l |_{y_i=0}{\mathcal {A}}_k |_{y_i=0}=\frac{\partial {\mathcal {A}}_k|_{y_i=0}}{\partial y_l}+{\mathcal {A}}_k|_{y_i=0}{\mathcal {A}}_l|_{y_i=0},\quad \quad k\ne i, ~l\ne i ,~ k\ne l. \end{aligned}$$

Hence, the following Pfaffian system is Frobenius integrable

$$\begin{aligned} \frac{\partial G}{\partial y_k}= \mathcal {A}_k|_{y_i=0} ~G~\equiv \left( \frac{B_k}{u_k-u_i}+\gamma _k\right) G ,\quad k\ne i. \end{aligned}$$

(9.5)

Using the chain rule as in (9.3), we receive (6.8)

$$\begin{aligned} \frac{\partial G}{\partial u_k} =\left( \frac{B_k}{u_k-u_i}+\gamma _k\right) G,\quad k\ne i, \quad \quad \quad \frac{\partial G}{\partial u_i}=-\sum _{k\ne i} \left( \frac{B_k}{u_k-u_i}+\gamma _k\right) G \end{aligned}$$

(9.6)

Notice that for both $\varphi (u)=B_i(u)$ and $\varphi (u)=G(u)$ we have

$$\begin{aligned} \sum _{k=1}^n \frac{\partial \varphi }{\partial u_k}=0\quad \Longrightarrow \quad \varphi (u)=\varphi (u_1-u_i,~\dots ~,u_n-u_i). \end{aligned}$$

(9.7)

We can take a solution G(u) which holomorphically reduces $B_i$ to Jordan form. Indeed

$$\begin{aligned} \hbox { for } k\ne i, \quad \frac{\partial }{\partial y_k} (G^{-1}B_iG)&= -G^{-1}\frac{\partial G}{\partial y_k} G^{-1}B_iG+G^{-1}\frac{\partial B_i}{\partial y_k} G + G^{-1} B_i \frac{\partial G}{\partial y_k} \\&\underset{(9.2),(9.5)}{=} -G^{-1}\mathcal {A}_k|_{y_i=0} B_i G \\&\quad + G^{-1}\bigl [\mathcal {A}_k|_{y_i=0},~B_i\bigr ]G +G^{-1}B_i \mathcal {A}_k|_{y_i=0} G \\&\quad \quad =0. \end{aligned}$$

Therefore, keeping into account (9.7), we see that $\mathcal {B}_i:=G^{-1}(u)B_i(u)G(u))$ is independent of u. Thus, there exists a constant matrix $\mathcal {G}$ such that $\mathcal {G}^{-1}\mathcal {B}_i \mathcal {G}$ is a constant Jordan form, and $G^{(i)}(u):=G(u) \mathcal {G}$ realizes the holomorphic “Jordanization”. The above arguments are standard, see for example [28]. $\square$

If the $B_i(u)$ are holomorphic on $\mathbb {D}(u^c)$ and the vanishing conditions (4.1) hold, the coefficients of the Pfaffian system (6.39) are holomorphic on $\mathbb {D}(u^c)$, so that $G^{(i)}(u)$ extends holomorphically there, and Corollary 9.1 holds on $\mathbb {D}(u^c)$.

Appendix B. Proof of Proposition 3.1

According to Theorem 2.1, system (1.1) is strongly isomonodromic in $\mathbb {D}(u^0)$ contained in a $\tau$-cell of $\mathbb {D}(u^c)$ if and only if (3.1) holds. In this case $G^{(0)}$ in (2.12) holomorphically reduces A(u) to constant Jordan form and satisfies

$$\begin{aligned} dG^{(0)}=\sum _{j=1}^n \omega _j(u) \mathrm{d}u_j ~G^{(0)}. \end{aligned}$$

(10.1)

Proof of Proposition 3.1

Suppose that (1.1) is strongly isomonodromic, so that (3.1) holds. Let $\mathcal {A}:=-A-I$, so that $E_k\mathcal {A}=B_k$, and (3.1) are rewritten as $\partial _i \mathcal {A}=[\omega _i(u),\mathcal {A}]$. We multiply these equations to the left by $E_k$, with $k\ne i$. We receive

$$\begin{aligned} E_k\partial _i \mathcal {A} =E_k[\omega _i(u),\mathcal {A}]. \end{aligned}$$

The l.h.s. is $E_k\partial _i \mathcal {A}=\partial _i B_k$. The r.h.s. is

$$\begin{aligned} E_k[ \omega _i,\mathcal {A}]=E_k \omega _i\mathcal {A}-E_k\mathcal {A} \omega _i=E_k \omega _i\mathcal {A}-B_k \omega _i = \bigl (E_k \omega _i\mathcal {A}- \omega _i B_k\bigr )+[\omega _i,B_k]. \end{aligned}$$

In conclusion

$$\begin{aligned} \partial _i B_k=\bigl (E_k \omega _i\mathcal {A}- \omega _i B_k\bigr )+[\omega _i,B_k],\quad i\ne k. \end{aligned}$$

The only terms we need to evaluate are

$$\begin{aligned}&E_k \omega _i\mathcal {A}- \omega _i B_k=E_k[F_1,E_i]\mathcal {A}-[F_1,E_i]B_k\\&\quad =E_kF_1E_i\mathcal {A}+E_iF_1B_k= E_kF_1E_i B_i+E_iF_1E_k B_k. \end{aligned}$$

In the second line we have used $E_iE_k=E_kE_i=E_iB_k=0$, for $i\ne k$, and $E_i^2=E_i$. Now, observe that $E_kF_1E_i$ has zero entries, except for the entry (k, i), which is $(F_1)_{ki}= (A)_{ki}/(u_i-u_k)$. This implies that

$$\begin{aligned} E_kF_1E_i B_i+E_iF_1E_k B_k= \frac{[B_i,B_k]}{u_i-u_k}. \end{aligned}$$

In conclusion, we have proved that (3.1) implies (3.4). On the other hand (3.4)–(3.5) are equivalent to the system given by (3.4) and the equations

$$\begin{aligned} \partial _i \sum _kB_k= [ \omega _i,\sum _k B_k],\quad i=1,\ldots ,n. \end{aligned}$$

which are exactly (3.1) if $B_k=E_k\mathcal {A}$. Finally, notice that (3.3), here with $\gamma _j=\omega _j$, is the integrability condition on $\mathbb {D}(u^0)$ of $dG=\sum _{j=1}^n \omega _j(u) \mathrm{d}u_j ~G$. On the other hand, it is a computation to see that (3.1) implies the the same conditions.

Conversely, let system (1.4) be strongly isomonodromic, so that the integrability conditions (3.3)–(3.5) hold. Firstly, we show that (3.4)–(3.5) imply a Pfaffian system for A of type (3.1). To this end, we sum (3.4) and (3.5):

$$\begin{aligned} \sum _{k=1}^n \partial _i B_k= \sum _{k\ne i} \frac{[B_i,B_k]}{u_i-u_k}- \sum _{k\ne i} \frac{[B_i,B_k]}{u_i-u_k}+\left[ \gamma _i, \sum _{k=1}^n B_k\right] =\left[ \gamma _i, \sum _{k=1}^n B_k\right] . \end{aligned}$$

Using $B_k=-E_k(A+I)$ and $\sum _k E_k=I$, the above becomes

$$\begin{aligned} \partial _i A=[\gamma _i,A],\quad i=1,\ldots ,n. \end{aligned}$$

(10.2)

Since $\gamma _1,\ldots ,\gamma _n$ satisfy (3.3), it is directly verified that (10.2) is Frobenius integrable. Secondly, we must show that we can choose

$$\begin{aligned} \gamma _j:=\omega _j= \left( \frac{A_{ab}(\delta _{aj}-\delta _{bj})}{u_a-u_b} \right) _{a,b=1}^n \quad \hbox { as in }(2.18). \end{aligned}$$

Substituting this choice into (3.3), we see that if (10.2) holds, in the form $\partial _i A=[\omega _i,A]$, then (3.3) are satisfied.²¹ Now, since (10.2) follows from (3.4) to (3.5) with matrices $B_k=-E_k(A+I)$, we conclude that (3.4)–(3.5) and the choice $\gamma _j=\omega _j$ guarantee that both (3.3) and (3.1) are satisfied. $\square$

The Schlesinger system can be used to show that there is a fundamental matrix solution $G^{(0)}$ of (10.1) that holomorphically reduces A to constant Jordan form on $\mathbb {D}(u^0)$. Since (3.3) is the integrability condition on $\mathbb {D}(u^0)$ of the linear Pfaffian system

$$\begin{aligned} dG=\sum _{j=1}^n \omega _j(u) \mathrm{d}u_j ~G. \end{aligned}$$

(10.3)

the latter admits holomorphic fundamental matrix solutions in $\mathbb {D}(u^0)$. Let G(u) be one of them and define

$$\begin{aligned} \widehat{B}_k:=G(u)^{-1} B_k G(u). \end{aligned}$$

(10.4)

By direct computation, using (10.3) and its integrability (3.3), it is verified that (3.4)–(3.5) (with $\gamma _j=\omega _j$) are equivalent to the normalized Schlesinger equations for the matrices $\widehat{B}_k$,

$$\begin{aligned} \partial _i\widehat{B}_k=\frac{[\widehat{B}_i,\widehat{B}_k]}{u_i-u_k}, \quad i\ne k;\quad \quad \partial _i\widehat{B}_i= -\sum _{k\ne i} \frac{[\widehat{B}_i,\widehat{B}_k]}{u_i-u_k}. \end{aligned}$$

The above equations imply that

$$\begin{aligned} \forall ~i=1,\ldots ,n,\quad \quad \partial _i \widehat{B}_\infty =0, \quad \quad \hbox { where } \widehat{B}_\infty := -\sum _{k=1}^n \widehat{B}_k \end{aligned}$$

Being $\widehat{B}_\infty$ constant, it can be put in constant Jordan form by a constant invertible matrix P, say $-\mathcal {J}=P^{-1} \widehat{B}_\infty P$. Since also G(u)P solves (10.3), we can choose from the beginning G(u) such that

$$\begin{aligned} G^{-1}(u) \left( \sum _{k=1}^n B_k(u)\right) G(u)=\mathcal {J} \quad \hbox { constant Jordan form}. \end{aligned}$$

(10.5)

Now, observe that $\sum _{k=1}^nE_k=I$, so that

$$\begin{aligned} \sum _{k=1}^n B_k=-\sum _{k=1}^n E_k (A+I)= -A-I. \end{aligned}$$

Thus, G(u) also puts A in constant Jordan form, so that²²

$$\begin{aligned} G(u)=G^{(0)}(u),\quad \quad \hbox { where } G^{(0)}\hbox { is in } (2.12). \end{aligned}$$

In particular, $G^{(0)}$ satisfies (10.1).

The second part of the statement of Proposition 2.3 (Prop. 19.2 of [13]) is now easily proved. Indeed, if $A(u)=G^{(0)}(u) J (G^{(0)})^{-1}$ holomorphically on $\mathbb {D}(u^c)$, where J is Jordan, then $G^{(0)}$ satisfies (10.1) on $\mathbb {D}(u^0)$ (and J is constant). Since $G^{(0)}(u)$ is holomorphic on $\mathbb {D}(u^c)$, the $\omega _j$ must be as well, so that the vanishing conditions (2.22) must hold. Conversely, if A is holomorphic on on $\mathbb {D}(u^0)$ and satisfies the vanishing conditions (2.22) (or, more weakly, if $dA=\sum _j[\omega _j ,A]\mathrm{d}u_j$ on $\mathbb {D}\backslash \Delta$, which automatically implies (2.22)—see Remark 2.1), then $dG=\sum _j\omega _j \mathrm{d}u_j ~G$ is integrable with holomorphic coefficients on $\mathbb {D}(u^0)$, and admits a fundamental matrix solution that can be chosen so that $(G^{(0)})^{-1}AG^{(0)}(u)=J$ (the proof is as done before on $\mathbb {D}(u^0)$).

Appendix C. The normal form (8.3)

We prove the expression (8.3) of Sect. 8, where it was sufficient to only consider the generic case of all $\lambda ^\prime _1,\ldots ,\lambda ^\prime _n ~\not \in \mathbb {Z}$ and no integer eigenvalues of A. A fundamental matrix solution in Levelt form at $\lambda =\lambda _1$ for system (8.1) has structure

$$\begin{aligned} \mathring{\Psi }(\lambda )=G^{(\varvec{p}_1)}\Bigl ( I+\sum _{l=1}^\infty \mathfrak {G}_l (\lambda -\lambda _1)^l \Bigr )(\lambda -\lambda _1)^{T^{(\varvec{p}_1)}}(\lambda -\lambda _1)^R, \end{aligned}$$

(11.1)

with

$$\begin{aligned} R=R_1+R_2+\cdots R_{\kappa },\quad \quad \kappa :=\max \{T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj} \hbox { integer}\}. \end{aligned}$$

where R is a nilpotent matrix with $R_{ij}\ne 0$ only if $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}$ is a positive integer. We prove that $R=0$ in our case. The formulae for $( \mathfrak {G}_l)_{ij}$ and $(R_l)_{ij}$ are obtained recursively by substituting the series into the differential system, and are as follows.

• If $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}= l$ (positive integer), $( \mathfrak {G}_l)_{ij}$ is arbitrary, and

  $$\begin{aligned} (R_l)_{ij}=\left( \sum _{p=1}^{l-1} (\mathfrak {D}_{l-p}\mathfrak {G}_l-\mathfrak {G}_l{R}_{l-p})+\mathfrak {D}_l\right) _{ij}, \end{aligned}$$

• If $T^{(\varvec{p}_1)}_{ii}-T^{(\varvec{p}_1)}_{jj}\ne l$ (positive integer)

  $$\begin{aligned} ( \mathfrak {G}_l)_{ij} =\frac{1}{T^{(\varvec{p}_1)}_{jj}-T^{(\varvec{p}_1)}_{ii}+l}\left( \sum _{p=1}^{l-1} (\mathfrak {D}_{l-p}\mathfrak {G}_l-\mathfrak {G}_l{R}_{l-p})+\mathfrak {D}_l \right) _{ij} \end{aligned}$$

The claim that $R=0$ follows from two facts. First, if we evaluate at $u=u^c$ the isomonodromic fundamental matrix solution (6.24), we receive a fundamental matrix solution of (1.4) at $u=u^c$,

$$\begin{aligned} \Psi ^{(\varvec{p}_1)}(\lambda ,u^c)=G^{(\varvec{p}_1)}\cdot U^{(\varvec{p}_1)}(\lambda ,u^c)\cdot (\lambda -\lambda _1)^{T^{(\varvec{p}_1)}}, \end{aligned}$$

(11.2)

which has $R=0$, because in the generic case here considered all $R^{(j)}=0$ in (6.24). The expression (11.2) belongs to the class of solutions (11.1).

The second fact is that other solutions in the class (11.1) may have different matrix exponents (see [27] and [13]; see also [14, 20] for the case of Frobenius manifolds), but if R corresponds to one solution, all the other solutions in the class can only have exponent

$$\begin{aligned} \widetilde{R}= \mathcal {D}^{-1} R \mathcal {D}, \end{aligned}$$

(11.3)

where $\mathcal {D}$ is an invertible matrix explained below. Now, since $R=0$ in (11.2), then by (11.3) all the other $\widetilde{R}=0$. This proves that (8.3) is the correct form.

Finally, we explain (11.3). System (1.4) at $u=u^c$ is holomorphically equivalent to “Birkhoff-normal forms”

$$\begin{aligned} \frac{\mathrm{d}\Psi }{\mathrm{d}\lambda }=\left( \frac{T^{(\varvec{p}_1)}}{\lambda -\lambda _1}+\sum _{l=1}^\kappa R_l(\lambda -\lambda _1)^l\right) \Psi \quad \hbox { and } \quad \frac{\mathrm{d}\widetilde{\Psi }}{\mathrm{d}\lambda }=\left( \frac{T^{(\varvec{p}_1)}}{\lambda -\lambda _1}+\sum _{l=1}^\kappa \widetilde{R}_l(\lambda -\lambda _1)^l\right) \widetilde{\Psi }, \end{aligned}$$

which are related to each other by a gauge transformations $\Psi =\mathcal {D}(\lambda ) \widetilde{\Psi }$, with $\mathcal {D}(\lambda )=\mathcal {D}_0(I+\mathcal {D}_0(\lambda -\lambda _1)+\cdots + \mathcal {D}_\kappa (\lambda -\lambda _1)^\kappa )$, where $\hbox {det}(\mathcal {D}_0)\ne 0$ and $[\mathcal {D}_0,T^{(\varvec{p}_1)}]=0$. Then, $\mathcal {D}:=\mathcal {D}_0(I+\mathcal {D}_0+\cdots + \mathcal {D}_\kappa )$.

Remark 11.1

In our case, the equations $R_l=0$, $l=1,2,\ldots ,\kappa$ are conditions on the entries of $A(u^c)$. The above discussion shows that, in the isomonodromic case, such conditions turn out to be automatically satisfied with the only vanishing assumption $(A(u^c))_{ab}=0$ for $u_a^c=u_b^c$. These conditions are equivalent to the conditions (4.24)–(4.25) of Proposition 4.2 in [13], and probably more convenient. We will not enter into the tedious verification of the equivalence.