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 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.