Mock theta functions and Appell–Lerch sums

Abstract

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) $g_{2}{(x,q)}$. The purpose of this paper is to consider the bilateral series for the universal mock theta function g3(x,q) $g_{3}{(x,q)}$. 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(ω;q) $B(\omega;q)$ for the third order mock theta function ω(q) $\omega(q)$ is established. The inner relationships between the two-group bilateral series of the fifth order mock theta functions are obtained as applications.

Keywords

• Mock theta functions
• Bilateral series
• Appell–Lerch sums