On the Difference of Partial Theta Functions

Abstract

Sums of the form $\sum _{n\ge 0} (-1)^nq^{(n-1)n/2} x^n$ 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 (Conjugate Bailey pairs from configuration sums and fractional-level string functions to Bailey’s lemma. Contemporary mathematics, American Mathematical Society, Providence, vol 297, pp 227–255, 2002). Finally, I show that Schilling–Warnnar (2002) and Warnaar (Proc Lond Math Soc 87:363–395, 2003) formulas are, in fact, equivalent.