Expansions of averaged truncations of basic hypergeometric series

TL;DR

This paper develops new expansion formulas for averaged truncations of basic hypergeometric series, extending previous work and providing tools for analyzing partition-related inequalities.

Contribution

It introduces novel expansion results for averaged truncations of key basic hypergeometric series, including those in the Jacobi triple product and ${}_5 ext{-} ext{phi}_5$ summation.

Findings

Derived new expansion formulas for hypergeometric series truncations

Recovered existing results as special cases

Established new inequalities related to partition counts

Abstract

Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving truncations of selected, more general, basic hypergeometric series. In particular, our expansions include new results for averaged truncations of the series appearing in the Jacobi triple product identity, the $q$-Gau{\ss} summation, and the very-well-poised ${}_5\phi_5$ summation. We show how special cases of our expansions can be used to recover various existing results. In addition, we establish new inequalities, such as one for a refinement of the number of partitions into three different colors.