Notes on q-analogues of a binomial congruence of Glaisher

TL;DR

This paper presents new q-analogues of a binomial congruence originally studied by Glaisher, addressing previous suggestions for perfect q-analogues and providing refined formulations.

Contribution

The authors construct and verify q-analogues that meet Andrews' criteria for perfection, advancing the understanding of binomial congruences in q-series.

Findings

Introduced new perfect q-analogues of Glaisher's binomial congruence

Validated the proposed q-analogues through mathematical proofs

Enhanced the theoretical framework of q-series and binomial congruences

Abstract

Andrews once gave $q$-analogues of a binomial congruence of Glaisher, and he suggested perfect $q$-analogues. In this note we give ones meeting the demand of Andrews.