# Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

## TL;DR

This paper explores identities of mock theta functions derived from bilateral basic hypergeometric series, utilizing Bailey's transformations to establish new summation formulas and analyze boundary behaviors at roots of unity.

## Contribution

It introduces new transformation and summation formulas for bilateral series related to mock theta functions using Bailey's identities.

## Key findings

- Derived new summation formulas for bilateral series associated with mock theta functions.
- Established explicit limits of mock theta functions minus theta functions at roots of unity.
- Connected bilateral hypergeometric series to the behavior of mock theta functions near roots of unity.

## Abstract

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2\psi_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. \\ New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say $\chi(q)$, and a root of unity in a certain class, say $\zeta$, that there is a theta function $\theta_{\chi}(q)$ such that \[ \lim_{q \to \zeta}(\chi(q) - \theta_{\chi}(q)) \] exists, as $q \to \zeta$ from within the unit circle.