# An extension of the Andrews-Warnaar partial theta function identity

## TL;DR

This paper extends the Andrews-Warnaar partial theta function identity using hypergeometric series transformations, unifying several classical results and establishing new identities for partial and false theta functions.

## Contribution

It introduces a new three-term identity for partial theta functions, extending previous work and connecting it with big q-Jacobi polynomials.

## Key findings

- Derived a three-term identity for partial theta functions
- Unified results by Ramanujan, Lovejoy, and Kim
- Established a relation between big q-Jacobi polynomials and theta functions

## Abstract

In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews-Warnaar partial theta function identity, and also unifies several results on partial theta functions due to Ramanujan, Lovejoy and Kim. We also establish a two-term version of the extension, which can be used to derive identities for partial and false theta functions. Finally, we present a relation between the big $q$-Jacobi polynomials and the Andrews-Warnaar partial theta function identity.