# Further Results on Vanishing Coefficients in Infinite Product Expansions

## TL;DR

This paper extends and refines previous results on when coefficients in certain infinite product series vanish, providing new cases and interpretations related to integer partitions.

## Contribution

It generalizes and simplifies existing theorems on coefficient vanishing in infinite products, covering new cases and offering partition interpretations.

## Key findings

- Identifies new vanishing coefficient cases in infinite product expansions.
- Provides simplified reformulations of earlier theorems by Alladi and Gordon.
- Offers interpretations of the results in terms of integer partitions.

## Abstract

We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if \begin{equation*} \frac{(q^{r-tk}, q^{mk-(r-tk)}; q^{mk})_\infty}{(q^r,q^{mk-r}; q^{mk})_\infty} =: \sum_{n=0}^\infty c_nq^n, \end{equation*} for certain integers $k$, $m$ $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms of integer partitions.