On a formula of the $q$-series $_{2k+4}\phi_{2k+3}$ and its applications

TL;DR

This paper derives new formulas for expressing complex $q$-series sums as linear combinations of infinite products, with applications to overpartition pairs and related combinatorial structures.

Contribution

It introduces a novel application of a specific $_{2k+4} ext{-} ext{phi}_{2k+3}$ formula to simplify and represent $q$-series sums in terms of infinite products.

Findings

Expressed $k$-tuple sums of $q$-series as linear combinations of product expressions.

Rewrote sums and double sums of $q$-series as linear combinations of infinite products.

Connected formulas to overpartition pairs and their combinatorial interpretations.

Abstract

In this paper we apply a formula of the very-well poised $_{2k+4}\phi_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy, qy^{-1};q)_\infty}$. As an application, we shall express a variety of sums and double sums of $q$-series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.