Mock theta functions and Appell–Lerch sums

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 [Formula: see text] . The purpose of this paper is to consider the bilateral series for the universal mock theta function [Formula: see text] . 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 [Formula: see text] for the third order mock theta function [Formula: see text] is established. The inner relationships between the two-group bilateral series of the fifth order mock theta functions are obtained as applications.