Transformation of Third Order Mock Theta Functions and New $q$-Series Identities

TL;DR

This paper explores transformations of third order mock theta functions, deriving new identities and providing a novel proof of Zwegers' theorem by connecting mock theta functions to Lerch sums and $q$-series.

Contribution

It introduces a new method to derive transformations of mock theta functions using Lerch sums and establishes new $q$-series identities, offering an alternative proof of Zwegers' theorem.

Findings

Derived transformations of third order mock theta functions.

Established new $q$-series identities.

Provided a new proof of Zwegers' theorem.

Abstract

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions $f(q)$ and $\omega$(q). Zwegers in 2000 built on Watson's techniques to complete these mock theta functions and connected them to real analytic modular forms. We show how to derive these transformations using Lerch sums. To show the equivalence of the results involves some new $q$-series identities thus resulting in a new proof of Zwegers' theorem.