A q-analog of Ljunggren's binomial congruence

TL;DR

This paper establishes a new q-analog of Ljunggren's classical binomial congruence, extending previous results and providing a deeper understanding of binomial coefficient congruences modulo prime powers.

Contribution

It introduces a novel q-analog of Ljunggren's congruence, generalizing earlier results and connecting multiple classical and q-analog binomial congruences.

Findings

Proves a q-analog of Ljunggren's binomial congruence for primes p ≥ 5.

Unifies and extends previous congruences by Babbage, Wolstenholme, Glaisher, and Clark.

Provides a new framework for binomial coefficient congruences in q-analog form.

Abstract

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that \[ \binom{a p}{b p} \equiv \binom{a}{b} \] modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.