# On the difference of partial theta functions

Alexander Berkovich

Arxiv ID: 0712.4087•Last updated: 6/8/2020

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.

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