On Andrews--Warnaar's identities of partial theta functions

TL;DR

This paper introduces a bivariate representation of partial theta functions that unifies existing identities and reveals new properties, leading to generalized identities and q-series transformations.

Contribution

It provides a novel bivariate framework for partial theta functions, unifying known identities and deriving new general forms and transformations.

Findings

Unified several famous identities for partial theta functions.

Established a general form of Warnaar's identity.

Derived a new q-series transformation related to Bailey pairs.

Abstract

In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such identities. As further applications, we establish a general form of Warnaar's identity and a general $q$--series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.