Mock theta functions and Appell\u2013Lerch sums

Bin Chen

doi:10.1186/s13660-018-1748-1

Abstract

Recently, Mortenson (Proc. Edinb. Math. Soc. 4:1–13, 2015) explored the bilateral series in terms of Appell–Lerch sums for the universal mock theta function g_{2}{(x,q)}. The purpose of this paper is to consider the bilateral series for the universal mock theta function g_{3}{(x,q)}. As a result, we present the bilateral series associated with the odd order mock theta functions in terms of Appell–Lerch sums. A very interesting congruence relationship of the bilateral series B(omega;q) for the third order mock theta function omega(q) is established. The inner relationships between the two-group bilateral series of the fifth order mock theta functions are obtained as applications.

Highlights

In 1920, the well-known mock theta functions were first introduced by Ramanujan in his last letter to Hardy [2, 3]
In 2002, Zwegers [6, 7] established the relationship between mock theta functions and real analytic vector-valued modular forms
The author and Zhou [21] found that the bilateral series B(f ; q) of the third order mock theta function f (q) is a mixed mock modular form of weight 1/2

Summary

Introduction

In 1920, the well-known mock theta functions were first introduced by Ramanujan in his last letter to Hardy [2, 3]. Let M(q) := n≥0 c(n; q) be a mock theta function, its associated bilateral series is defined as The author and Zhou [21] found that the bilateral series B(f ; q) of the third order mock theta function f (q) is a mixed mock modular form of weight 1/2.

Objectives

Results

Conclusion