Mock theta functions and Appell–Lerch sums

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 g2(x,q). The purpose of this paper is to consider the bilateral series for the universal mock theta function g3(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(ω;q) for the third order mock theta function ω(q) is established. The inner relationships between the two-group bilateral series of the fifth order mock theta functions are obtained as applications.