The $p$-Dissection of a Product of Quintuple Products

TL;DR

This paper derives explicit formulas for the p-dissection of a product of quintuple products, analyzes coefficient sign patterns in related series, and explores combinatorial applications, advancing understanding of these specialized q-series identities.

Contribution

It provides new explicit p-dissection formulas for a product involving the quintuple product identity and analyzes coefficient sign patterns, which were previously unknown.

Findings

Explicit p-dissection formulas for the product Q(z,q)

Identification of sign patterns in Taylor series coefficients

Applications to combinatorial identities and problems

Abstract

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given.