Some new mock theta functions

Abstract

Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some q-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double sums, we provide the Hecke-type double sums for some third order mock theta functions, and establish the relations between these functions and some classical mock theta functions.

Introduction

Mock theta functions were first introduced by Ramanujan [30] in his last letter to Hardy. He found that these functions have certain asymptotic properties as q approaches a root of unity, which are similar to theta functions, but they are not really theta functions. Historically, mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In these years, it has received a great deal of attention to seek new mock theta functions and find different forms for mock theta functions. In this paper, we find the following two-parameter mock theta functions:

$$U_1(x,q):=(1+x)(1+xq)(1+x^{-1}q)\sum _{n=0}^{\infty }\frac{{(-1;q)}_{2n}{q}^{3n}}{{(xq,x^{-1}q;q^2)}_{n+1}},U_2(x,q):=(1+q)(1+xq)(1+x^{-1}q)\sum _{n=0}^{\infty }\frac{{(-1,-q^3;q^2)}_n{q}^n}{{(xq,x^{-1}q;q^2)}_{n+1}},U_3(x,q):=(1+q)(1-xq)(1-x^{-1}q)\sum _{n=0}^{\infty }\frac{{(q,-q^3;q^2)}_n{(-1)}^n{q}^{2n+1}}{{(x{q}^2,x^{-1}{q}^2;q^2)}_{n+1}}.$$

In addition, by finding the following functions:

$$E_1(q):=\sum _{n=0}^{\infty }\frac{q^{n^2}}{{(q;q^2)}_{n+1}},$$

$$E_2(q):=\sum _{n=0}^{\infty }\frac{(1-q)q^{n^2+2n}}{{(-q^2;q^2)}_n},$$

$$E_3(q):=\sum _{n=0}^{\infty }\frac{(1-q)q^{n^2+n}}{{(-q;q^2)}_n},$$

we establish some relations between these functions and some third order mock theta functions, and then provide the Hecke-type double sums for these third order mock theta functions.

Here and throughout the paper, we use the standard q-series notation [16]. Let q denote a complex number with $|q|<1$. Then for positive integers n and m,

$${(a;q)}_0:=1,{(a;q)}_n:=\prod _{k=0}^{n-1}(1-a{q}^k),{(a;q)}_{-n}:=\frac{q^{\left(\begin{array}{c}n\\ 2\end{array}\right)}{(-q/a)}^n}{{(q/a;q)}_n},{(a;q)}_{\infty }:=\prod _{k=0}^{\infty }(1-a{q}^k),{(a_1,a_2,\cdots ,a_m;q)}_{\infty }:={(a_1;q)}_{\infty }{(a_2;q)}_{\infty }\cdots {(a_m;q)}_{\infty }.$$

The (unilateral) basic hypergeometric series ${}_r\varphi _s$ is defined as

$${}_r\varphi _s(\begin{array}{c}\hfill a_1,a_2,\dots ,a_r\hfill \\ \hfill b_1,\dots ,b_s\hfill \end{array};q,x)=\sum _{n=0}^{\infty }\frac{{(a_1,a_2,\dots ,a_r;q)}_n}{{(q,b_1,\dots ,b_s;q)}_n}{\left({(-1)}^n{q}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}\right)}^{1+s-r}{x}^n.$$

Jacobi's triple product identity is given by

$$j(x;q):={(x;q)}_{\infty }{(q/x;q)}_{\infty }{(q;q)}_{\infty }=\sum _{n=-\infty }^{\infty }{(-1)}^n{q}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}{x}^n.$$

Let a and m be integers with m positive. Then define

$$J_{a,m}:=j(q^a;q^m),{\bar{J}}_{a,m}:=j(-q^a;q^m),J_m:=J_{m,3m}=\prod _{i=1}^{\infty }(1-q^{mi}).$$

Ramanujan listed 17 mock theta functions which were assigned orders 3, 5, and 7 in the last letter to Hardy. The following two third order mock theta functions are included in the list.

$$\varphi (q)=\sum _{n=0}^{\infty }\frac{q^{n^2}}{{(-q^2;q^2)}_n},$$

$$\psi (q)=\sum _{n=1}^{\infty }\frac{q^{n^2}}{{(q;q^2)}_n}.$$

In 1936, Waston [32] found another three third order mock theta functions. For example,

$$\nu (q)=\sum _{n=0}^{\infty }\frac{q^{n^2+n}}{{(-q;q^2)}_{n+1}}.$$

Then Waston [33] proved some linear relations for the fifth order mock theta functions given by Ramanujan. In 1981, by means of Bailey chains, Andrews [2] established the Hecke-type double sums for the fifth and seventh order mock theta functions. Hecke-type double sums have the following form:

$$\sum _{(m,n)\in D}{(-1)}^{H(m,n)}{q}^{Q(m,n)+L(m,n)},$$

where $H(m,n)$ and $L(m,n)$ are linear forms, $Q(m,n)$ is an indefinite quadratic form, and D is some subset of $\mathbb{Z}\times \mathbb{Z}$ such that $Q(m,n)\ge 0$ for all $(m,n)\in D$. For example, Andrews gave the following Hecke-type identities:

$$J_1{f}_0(q)=\sum _{n=0}^{\infty }(1-q^{4n+2})q^{\frac{5n^2+n}{2}}\sum _{j=-n}^n{(-1)}^j{q}^{-j^2},$$

$$J_1{f}_1(q)=\sum _{n=0}^{\infty }(1-q^{2n+1})q^{\frac{5n^2+3n}{2}}\sum _{j=-n}^n{(-1)}^j{q}^{-j^2},$$

where the fifth order mock theta functions $f_0(q)$ and $f_1(q)$ are stated as

$$f_0(q)=\sum _{n=0}^{\infty }\frac{q^{n^2}}{{(-q;q)}_n}\text{and}{f}_1(q)=\sum _{n=0}^{\infty }\frac{q^{n^2+n}}{{(-q;q)}_n}.$$

Subsequently, Hecke-type double sums were widely used to prove identities related to mock theta functions. Based on (1.8) and (1.9), Hickerson [21] introduced the universal mock theta function $g(x,q)$,

$$g(x,q)=x^{-1}(-1+\sum _{n=0}^{\infty }\frac{q^{n^2}}{{(x;q)}_{n+1}{(x^{-1}q;q)}_n})=\sum _{n=0}^{\infty }\frac{q^{n(n+1)}}{{(x,x^{-1}q;q)}_{n+1}},$$

and then proved

$$f_0(q)=\frac{J_{5,10}{J}_{2,5}}{J_1}-2q^{2}g(q^2,q^{10}),f_1(q)=\frac{J_{5,10}{J}_{4,5}}{J_1}-2q^{3}g(q^4,q^{10}).$$

The above identities are called as “mock theta conjectures” which express the fifth order mock theta functions in terms of $g(x,q)$ and theta functions. It is customary to refer to the analogous identities involving other mock theta functions as mock theta conjectures. Later, based on $g(x,q)$ and the Hecke-type double sums given by Andrews [2], three mock theta conjectures related to the seventh order mock theta functions were proved in [20]. Furthermore, McIntosh [28] proved the mock theta conjectures for the third order mock theta functions. For example,

$$\varphi (q)=-2q{g}_3(q,q^4)+\frac{J_2^7}{J_1^3{J}_4^3},$$

where he used $g_3(x,q)$ to denote $g(x,q)$. In 2012, in view of q-orthogonal polynomials, Andrews [3] established some Hecke-type double sums related to some known and new third order mock theta functions. For example,

$$1+\psi (q)=\frac{1}{J_1}\sum _{n=0}^{\infty }(1-q^{6n+6}){(-1)}^n{q}^{2n^2+n}\sum _{j=0}^n{q}^{-\left(\begin{array}{c}j+1\\ 2\end{array}\right)}.$$

In [22], Hickerson and Mortenson gave the following definition for a special type of Hecke-type double sums.

Definition 1.1

Let x, $y\in {\mathbb{C}}^{⁎}:=\mathbb{C}﹨\{0\}$ and define sg(r) $:=1$ for $r\ge 0$ and sg(r) $:=-1$ for $r<0$. Then

$$f_{a,b,c}(x,y,q):=\sum _{sg(r)=sg(s)}sg(r){(-1)}^{r+s}{x}^r{y}^s{q}^{a\left(\begin{array}{c}r\\ 2\end{array}\right)+brs+c\left(\begin{array}{c}s\\ 2\end{array}\right)}.$$

It is clear that

$$f_{a,b,a}(x,y,q)=f_{a,b,a}(y,x,q).$$

In [29], Mortenson obtained different Hecke-type double sums for some third order mock theta functions. For example,

$$1+2\psi (q)=\frac{1}{J_1}\sum _{n=0}^{\infty }(1+q^{2n+1}){(-1)}^n{q}^{2n^2+n}\sum _{j=-n}^n{q}^{-\left(\begin{array}{c}j+1\\ 2\end{array}\right)},$$

which can be rewritten as

$$\psi (q)+1=\frac{1}{J_1}{f}_{3,5,3}(q^2,q^3,q).$$

For the even order mock theta functions, in view of Bailey's Lemma, Andrews and Hickerson [6] established the Hecke-type double sums for the sixth order mock theta functions, and proved the linear relations for them given by Ramanujan. In [7], Berndt and Chan found two more sixth order mock theta functions. Furthermore, by means of some classical q-series identities, Lovejoy [25] proved some relations for the sixth order mock theta functions. In view of the half-shifted method, Gordon and McIntosh [17] found eight eighth order mock theta functions. Later, McIntosh [26] studied three second order mock theta functions. In [14], the authors provided the Hecke-type double sums for the second and eighth order mock theta functions. For the tenth order mock theta functions appearing in the Lost Notebook [31], Choi [10], [11], [12], [13] established the Hecke-type double sums for these functions, and proved eight linear relations given by Ramanujan.

In [19], it shows that the mock theta functions with odd order can be expressed by the universal mock theta function $g_3(x,q)$. Similarly, the mock theta functions with even order are related to another universal mock theta function $g_2(x,q)$,

$$g_2(x,q)=\sum _{n=0}^{\infty }\frac{{(-q;q)}_n{q}^{n(n+1)/2}}{{(x,x^{-1}q;q)}_{n+1}}.$$

In 2014, Hickerson and Mortenson [22] provided the following definition of Appell–Lerch sums.

Definition 1.2

Let x, $z\in {\mathbb{C}}^{⁎}:=\mathbb{C}﹨\{0\}$ with neither z nor xz an integral power of q. Then

$$m(x,q,z):=\frac{1}{j(z;q)}\sum _{r=-\infty }^{\infty }\frac{{(-1)}^r{q}^{\left(\begin{array}{c}r\\ 2\end{array}\right)}{z}^r}{1-q^{r-1}xz}.$$

Changing r to $r+1$ in the above series gives another useful form for $m(x,q,z)$ [22, Eq. (3.1)]:

$$m(x,q,z)=\frac{-z}{j(z;q)}\sum _{r=-\infty }^{\infty }\frac{{(-1)}^r{q}^{\left(\begin{array}{c}r+1\\ 2\end{array}\right)}{z}^r}{1-q^{r}xz}.$$

Hickerson and Mortenson showed that all of Ramanujan's classical mock theta functions as well as those of Gordon and McIntosh can be expressed in terms of Appell–Lerch sums. In light of Appell-Lerch sums, results for mock theta functions are much more accessible than they were 20 years ago. For more on mock theta functions, one can see [1], [4], [5], [8], [15], [18], [27].

In this paper, we establish the Appell–Lerch sums for $U_1(x,q)$, $U_2(x,q)$, and $U_3(x,q)$.

Theorem 1.3

We have

$$U_1(x,q)=-2x{q}^{-1}(1-q)m(x,q^2,q)-2(1+x)m(-q,q^2,q).$$

Theorem 1.4

We have

$$U_2(x,q)=-2(1+x)m(x,q^2,q)+2(1-q)m(-q,q^2,q).$$

Theorem 1.5

We have

$$U_3(x,q)=2(1-x)m(-x,q^2,-1)-2(1-q)m(-q,q^2,-1).$$

Notice that by using Corollary 3.2 and Theorem 3.3 in [22], we can evaluate the following terms.

$$m(-q,q^2,q)=m(-q,q^2,-1)=\frac{1}{2}.$$

In addition, in view of some q-series identities given by Liu [23], [24] and Chen and Wang [9], we obtain the following Hecke-type double sums related to $E_1(q)$, $E_2(q)$, and $E_3(q)$.

Theorem 1.6

We have

$$E_1(q)-1=-\frac{2q^5}{J_{1,4}}{f}_{1,2,1}(q^9,q^{11},q^4).$$

Theorem 1.7

We have

$$E_2(q)-2=-\frac{1}{J_{1,4}}{f}_{1,2,1}(iq,-iq,q).$$

Theorem 1.8

We have

$$E_3(q)-1=-\frac{q}{J_{2,4}}{f}_{1,2,1}(i{q}^{\frac{3}{2}},-i{q}^{\frac{3}{2}},q).$$

Meanwhile, we provide some relations between these functions and the third order mock theta functions.

Theorem 1.9

We have

$$E_1(q)-2\psi (q)=1,$$

$$E_2(q)+\varphi (q)=2,E_3(q)+q\nu (q)=1.$$

Then according to Theorem 1.6, Theorem 1.9, we derive the following Hecke-type double sums for $\psi (q)$, $\varphi (q)$, and $\nu (q)$.

Corollary 1.10

We have

$$\psi (q)=-\frac{q^5}{J_{1,4}}{f}_{1,2,1}(q^9,q^{11},q^4),\varphi (q)=\frac{1}{J_{1,4}}{f}_{1,2,1}(iq,-iq,q),\nu (q)=\frac{1}{J_{2,4}}{f}_{1,2,1}(i{q}^{\frac{3}{2}},-i{q}^{\frac{3}{2}},q).$$

Notice that the Hecke-type double sum of $\nu (q)$ [9, Equation (6.30)] can be converted to (1.14).

This paper is organized as follows. In Section 2, we state some lemmas which are used to prove the main theorems. In Section 3, we prove Theorem 1.3, Theorem 1.5. In Section 4, we prove Theorem 1.6, Theorem 1.9.