Vanishing coefficients in some $q$-series expansions

TL;DR

This paper investigates vanishing coefficients in certain $q$-series expansions, extending recent work by proving new results about the zeros of coefficients in variants of these series.

Contribution

It introduces new variants of $q$-series and establishes vanishing coefficient results analogous to recent findings, expanding understanding of their arithmetic properties.

Findings

Proves that specific coefficients vanish for certain arithmetic progressions.

Extends previous results to new $q$-series variants.

Provides explicit conditions for coefficient vanishing.

Abstract

Motivated by the recent work of Hirschhorn on vanishing coefficients of the arithmetic progressions in certain $q$-series expansions, we study some variants of these $q$-series and prove some comparable results. For instance, let \begin{align*} (-q,-q^{4};q^{5})_{\infty}^{2}(q^{4},q^{6};q^{10})_{\infty}=\sum_{n=0}^{\infty}a_{1}(n)q^{n}, \end{align*} then \begin{align*} a_{1}(5n+3)=0. \end{align*}