Representations of mock theta functions

Abstract

Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters a and b. Specializing the choices of (a, b), we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same (a, b)-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks m(x, q, z) and f a,b,c (x, y, q) introduced by Hickerson and Mortenson, as well as m(x, q, z) and f a,b,c (x, y, q) introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.

Hecke-type double sums and mock theta functions

“…In this paper, with the aid of some q-series identities given by Liu [L131, L132] and Chen and Wang [CW20], we find Hecke-type double sums for some new mock theta functions. The main theorems are as follows.…”

Expansion formulas for multiple basic hypergeometric series over root systems

“…In particular, Liu extended Rogers' non-terminating very-well-poised 6 φ 5 summation formula, Watson's transformation formula, and gave an alternate approach to the orthogonality of the Askey-Wilson polynomials; see Liu [12,14,13,15] and Liu and Zheng [16]. These have been shown to be useful in number-theoretic contexts too; in particular, Chan and Liu [3], Wang and Yee [28], Wang [27] and Chen and Wang [4] have used them to prove Hecketype identities, and identities for false theta and mock theta functions. All this work relies on three expansion formulas of Liu.…”

Parity of coefficients of mock theta functions

“…For example, f (3) (q) := ∞ n=0 q n 2 (−q; q) 2 n and f (5) 0 (q) := ∞ n=0 q n 2 (−q; q) n (1.1) are two mock theta functions of orders 3 and 5, respectively. Here we follow [14] to add a superscript (n) to indicate that a mock theta function is of order n. We also adopt the customary q-series notation:…”

“…For a more comprehensive historical background on mock theta functions, see the survey of Gordon and McIntosh [27], the paper of Hickerson and Mortenson [30] or the recent book of Andrews and Berndt [2]. We remark that in a recent work [14], Chen and the author provided a unified method for establishing Appell-Lerch and Hecke-type series representations for mock theta functions of orders 2, 3, 5, 6 and 8.…”

“…Given a mock theta function g(q), our aim is to find its type modulo 2 and give characterizations for n such that c(g; n) is odd. Following the notations of mock theta functions in [14], we can state our main result as follows.…”