We prove a new addition theorem for cylinder functions. The value of an arbitrary zero order cylinder function for an argument x−y with complex-valued x and y is expressed by an infinite series whose summands involve the product of cylinder functions, one of these functions taken at x and the other at y. Other as in the hitherto known addition theorems for cylinder functions, the summands of this series

The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib–Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter

(2021). Special issue OPSFA15: orthogonal polynomials, special functions and applications. Integral Transforms and Special Functions: Vol. 32, Orthogonal Polynomials, Special Functions, and Applications (OPSFA15), pp. 333-335.

The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given.

We investigate interlacing properties of zeros of Laguerre polynomials Ln(α)(x) and Ln+1(α+k)(x), α>−1, where n∈N and k∈{1,2}. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp t-interval within which the zeros of two equal degree Laguerre polynomials Ln(α)(x) and Ln(α+t)(x) are interlacing for every n∈N and each α>−1 is 0−1 is 0≤t≤ 2, [Driver K

ABSTRACT We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as well as the Rodrigues formula for these polynomials are obtained. The difference equation of order r + 1 is studied. The connection via limit relation

We investigate asymptotic behaviour of polynomials pnω(z) satisfying varying non-Hermitian orthogonality relations ∫−11xkpnω(x)h(x)eiωxdx=0,k∈{0,…,n−1},where h(x)=h∗(x)(1−x)α(1+x)β, ω=λn, λ≥ 0 and h(x) is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials ( α=β=0) introduced in Asheim et al. [A Gaussian quadrature

Abstract The 3-term recurrence relation for Hermite polynomials was recently generalized to a recurrence relation for Wronskians of Hermite polynomials. In this note, a similar generalization for Laguerre polynomials is obtained.

Let (μ1,μ2) be a coherent pair of Jacobi measures. In most cases, we obtain convergence and boundedness of the Fourier series for Sobolev polynomials with respect to this kind of measures.

ABSTRACT We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight ω(x;t,λ)=|x|2λ+1exp−x6+tx2,x∈R, with parameters λ>−1 and t∈R. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions 1F2(a1;b1,b2;z). We derive a nonlinear discrete as well as a system of differential

ABSTRACT In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé

ABSTRACT We study the logarithmic asymptotic of multiple orthogonal polynomials arising in a mixed-type Hermite–Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite–Padé approximants.

Motivated by arithmetic properties of partition numbers p(n), our goal is to find algorithmically a Ramanujan type identity of the form ∑n=0∞p(11n+6)qn=R, where R is a polynomial in products of the form eα:=∏n=1∞(1−q11αn) with α=0,1,2. To this end we multiply the left side by an appropriate factor such the result is a modular function for Γ0(121) having only poles at infinity. It turns out that polynomials

ABSTRACT We consider several special functions defined by matrix integrals on Hermitian matrix space, namely, analogues of Gauss hypergeomeric, Kummer's confluent hypergeometric, Bessel, Hermite-Weber and Airy function. We discuss the relation of these functions of matrix integral type with some semi-classical orthogonal polynomials and with the polynomial solutions of quantum Painlevé equations.

ABSTRACT As an extension of the Askey-scheme, recently, it has been introduced the so-called d-Askey-scheme, which is a table describing families of hypergeometric d-orthogonal polynomials. In this paper, we derive limit relations linking between all levels of the 2-Askey-scheme (Figure 1). To this end, we use three methods. The first one is based on a recurrence relation. The second one uses difference

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix whose entries are explicitly computable in many applications. We apply this result to the Toeplitz operator with symbol zn+c|z|s acting on certain weighted Bergman

ABSTRACT The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter α>−1 and, in general, two point masses N≥0, S>0 at the origin involving functions and their first derivatives. Moreover we consider the Jacobi-Sobolev polynomials on the interval [−1,1] with Jacobi parameters α,β>−1 and one Sobolev point

In this paper, we study the sequences of matrix Cauchy biorthogonal polynomials. We will focus on the algebraic aspects of the problem, finding a connection with a kind of matrix mixed-type Hermite–Padé approximation problem and a Riemann–Hilbert problem. From here, we deduce some properties of these families of matrix Cauchy biorthogonal polynomials.

ABSTRACT The wave equation ∂tt−c2Δxu(x,t)=e−tf(x,t) in the cone {(x,t):∥x∥≤t,x∈Rd,t∈R+} is shown to have a unique solution if u and its partial derivatives in x are in L2(e−t) on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the

Solving a moment problem means, roughly speaking, to find a spectral representation of a given data. This is usually understood as a typical inverse spectral problem, also because of the traditional connection between moments and operators; the latter is the object of special concern in the present paper.

WKB theoretic transformation to the boosted Airy equation is discussed. The formulae for the discontinuities of the Borel transforms of the WKB solutions to a differential equation which can be transformed to the boosted Airy equation is given.

In my joint papers with Iwaki and Koike, we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper, we discuss its generalization to the case of the hypergeometric differential equation

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results

In this work, the quadratic-phase Fourier transform (QPFT) on Schwartz space is defined and its necessary results are derived, including adjoint formula, Parseval identity, and continuity property. Furthermore, the continuity property on tempered distribution space is discussed, and also an example of the QPFT is provided. The present analysis is applied to develop a new class of pseudo-differential

An investigation into families of integrals containing powers of the arctanh and log functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals. In another special case we prove that the corresponding log-tanh integral can be represented as a linear combination of the product of zeta functions and the Dirichlet eta function

For the Lauricella hypergeometric function FD(N) with an arbitrary number of variables z1,…,zN, we construct formulas for analytic continuation into the vicinity of hyperplanes {zj=zl} and their intersections providing that all variables are close to unit. Such formulas represent the function FD(N) near the point (1,…,1) as linear combinations of N–multiple hypergeometric series that are solutions

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright

Consider a function with a power series expansion and roots {xn,1≤n<∞}. We give two methods for obtaining the sums of powers of these roots, Sk=∑n=1∞xnk.The first is by expressing Girard's formula in terms of Bell polynomials. Explicit expressions are given for Sk for k≤10. The second method is through generating functions.

In this paper, some elementary and useful terminologies associated to Lebedev–Skalskaya transform (LS-transform) are notified and then estimates for Rκ12+iτ(x), Fκ12+iτ(x), translation and convolution operators associated to LS-transform are obtained. The continuity of LS-transforms, translation and convolution operators on functions spaces Sα,k, Hα,k and Gα,k are discussed. The continuity of pseudo-differential

Some new relations for the Horn function H5(a,b;c;w,z) and confluent Horn function H5(c)(a;c;w,z) are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

In this paper, we give explicit evaluation for some integrals involving polylogarithm functions of types ∫0xtmLip(t)dt and ∫0xlogm⁡(t)Lip(t)dt. Some more integrals involving the logarithm function will also be derived.

Some inherited properties of exceptional Jacobi polynomials are derived and as an application it is shown that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense.

We consider the Lauricella hypergeometric function FD(N), depending on N≥2 variables z1,…,zN, and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the hyperplanes {zj=zl}. It is assumed that all N variables are large in modulo. This formulas represent the function FD(N) outside of the unit polydisk in the form of linear combinations of other

In this paper, we prove some properties of a class of integral operators defined in Lα2(R+n), and we characterize a relation between this class of operators and multidimensional Fourier-Bessel transform Fα associated with the differential operator Δα, Δα=∑i=1n∂2∂xi2+2αi+1xi∂∂xi,αi>−12. We also construct a class of integral operators which commute with multidimensional Fourier-Bessel transform Fα.

ABSTRACT We establish a spherical mean value formula for the iterated Dunkl-Helmoltz operator, thus generalizing a result of Mejjaoli and Trimèche. We then give an application to distributions with Dunkl transform supported by the unit sphere.

A generalization of the ultra-hyperbolic time-fractional diffusion-wave equation introduced by the author in Dorrego [The Mittag–Leffler function and its application to the ultra-hyperbolic time-fractional diffusion-wave equation. Integral Transforms Spec. Funct. 2016;27(5):392–404] is presented and studied. The results obtained here extend the previous to the whole space Rn. The problem is solved

In this paper, we consider the normalized Bessel function of index α>−12, we find an integral representation of the term xnjα+n(x)jα(y). This allows us to establish a product formula for the generalized Hankel function Bλκ,n on R. Bλκ,n is the kernel of the integral transform Fκ,n arising from the Dunkl theory. Indeed we show that Bλκ,n(x)Bλκ,n(y) can be expressed as an integral in terms of Bλκ,n(z)

In this paper, we investigate the fractional power (−Δν)γ/2, 0<γ<2, of the Bessel operator in the form Δν:=d2dx2+2ν+1xddx,ν>−1/2. Our method uses a canonical representation for generalized Laplacian related to the structures of the Bessel–Kingman hypergroup. As a direct application, we solve the space-fractional diffusion equation.

In this paper, we consider the third-order differential equation of the quadratic products of Hermite functions and present the integral representations of products of Hermite functions (Lebedev identities) in terms of the confluent hypergeometric functions. Manipulating the integrands of these identities with various integral representations lead us to get new representations for the products of Hermite

We derive relations for a certain class of terminating 4F3(4) hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group S3. We further study relations for terminating 3F2(4) series that fall under two families. By using a series reversal, we examine the corresponding terminating 4F3(1/4) and 3F2(1/4) series relations

Discrete analogues of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function 2F1(a+in/2,a−in/2;c;−x2),x>0,n∈N,a,c>0,i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.

We investigate the interplay between the Caputo fractional operators D±1/2 and Fourier–Legendre expansions of hypergeometric functions, resulting in transformation formulas for q+1Fq(z) series with quarter-integer parameters and in explicit evaluations of many series, including: ∑n≥0((12)nn!)2(−1)n2n+1,∑n≥0((12)nn!)214n+3,∑n≥0((12)nn!)31n+1,∑n≥0((12)nn!)31(n+1)2,∑n≥0((12)nn!)2H2n2n(n+1),∑n≥0((12)nn

Discrete analogues of the index transforms with squares of Bessel functions of the first and second kind Jν(z),Yν(z) are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

ABSTRACT The purpose of this work is to prove analogues of the classical Titchmarsh theorem and Younis' theorem associated with the Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition of a certain order in suitable weighted spaces Lp([0,+∞)), 1

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag–Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible.

ABSTRACT In this work, we will study a Bessel potential spaces Wαs,p(R+n), ( s∈R, 1≤p<+∞, α=(α1,…,αn)∈Rn, α1>−12,…,αn>−12) associated with the Poly-axially operator Δα and their properties. We define the Sobolev-type spaces Eαm,p(R+n) (m∈N, 0∈N, 1≤p<+∞) and we prove an analogue of Calderón's theorem for Fourier–Bessel transform. We define the nonlinear Bessel potentials associated with a measure ν

A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms which are yet to be named. The assumptions allow for certain properties to exist for these integral transforms, such as the shifting and convolution properties. Further conditions are presented which allow for

The coupled fractional Fourier transform is a two-dimensional fractional Fourier transform Fα,β that depends on two angles α,β that are coupled in such a way that the transform parameters are γ=(α+β)/2 and η=(α−β)/2. It generalizes the two-dimensional Fourier transform and it serves as useful tool in some applications in optics and signal processing. In this article we derive new properties of the

Thermal equilibrium of a two-dimensional model of mobile point-like charges with pair logarithmic interactions, immersed in a fixed neutralising background, is exactly solvable at a special coupling constant. Assuming gauge invariance of the statistical mechanics and the requirement of local charge neutrality in the bulk regime of the system, the exact solution implies explicit expressions for inverse

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence

It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new modified 2D-Laguerre polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new modified 2D-Laguerre polynomials and discuss the operational links with various known polynomials

In this paper, we consider the ∗w-product on function space Ca,b[0,T] introduced in Chang et al. [A new approach method to obtain the L1 generalized analytic Fourier–Feynman transform. Integr Transf Spec Funct. 2018;29:745–760]. We then obtain various integration formulas involving the ∗w-product and the first variation. Finally, we have relationships between the ∗w-product and the generalized integral

The exponential integral function Ei(x) is given as an indefinite integral of an elementary expression. This allows a second-order linear differential equation for the function to be constructed, which is of conventional form. A limitless number of differential equations can be derived from the original by elementary transformations, and many integrals are given by applying the method of fragments

ABSTRACT New asymptotic expansions are derived of the Kummer functions M(a,b,z) and U(a,b+1,z) for large positive values of a and b, with z fixed. For both functions we consider b/a≤1 and b/a≥1, with special attention for the case a∼b. We use a uniform method to handle all cases of these parameters.

ABSTRACT For fixed reals c>0, a>0 and α > − 1 2 , the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions are the eigenfunctions of the finite Hankel transform operator, denoted by H c α , which is the integral operator defined on L 2 ( 0 , 1 ) with kernel H c α ( x , y ) = c x y J α ( c x y ) . Also, they are the eigenfunctions of the positive, self-adjoint compact integral

ABSTRACT Sufficient conditions are obtained for which localized fractional derivatives of the Riemann–Liouville and Marshaud type can be represented as localized fractional derivatives of Djrbashian–Caputo type. The compositional properties of localized fractional derivatives and ordinary derivatives are proved. The localized derivatives for the exponential, trigonometric, power and Weierstrass's functions

Some new relations for the Horn function H 4 ( a , b ; c , c ′ ; w , z ) and confluent Horn function H 7 ( c ) ( a ; c , c ′ ; w , z ) are obtained including differentiation and integration formulas, series and for special values of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

Discrete analogs of the Lebedev–Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels R e   K α + i n ( x ) , I m   K α + i n ( x ) , x > 0 , n ∈ N , | α | < 1 ,   i is the imaginary unit and K ν ( z ) is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals

Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{\alpha+in}(x), {\rm Im} K_{\alpha+in}(x), x >0, n \in \mathbb{N}, |\alpha | < 1,\ i $ is the imaginary unit and $K_\nu(z)$ is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of