On the difference of partial theta functions

TL;DR

This paper explores identities involving partial theta functions, providing new proofs and establishing equivalences between known formulas, thereby deepening understanding of their interrelations and extending classical results.

Contribution

It offers a new elegant proof of a difference formula for partial theta functions and demonstrates the equivalence of two previously known identities.

Findings

New proof for the difference of two partial theta series

Established the equivalence of Schilling-Warnaar and Warnaar formulas

Connected the Andrews-Warnaar identity with Gasper-Rahman product formula

Abstract

Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta functions. Subsequently, Andrews and Warnaar established a similar result for the product of two partial theta functions. In this note, I discuss the relation between the Andrews-Warnaar identity and the (1986) product formula due to Gasper and Rahman. I employ nonterminating extension of Sears-Carlitz transformation for 3\phi_2 to provide a new elegant proof for a companion identity for the difference of two partial theta series. This difference formula first appeared in the work of Schilling-Warnaar (2002). Finally, I show that Schilling-Warnnar (2002) and Warnaar (2003) formulas are, in fact, equivalent.