A q-analog of Ljunggren's binomial congruence

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 $p\ge 5$. 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.

6 pages, to be published in the proceedings of FPSAC 2011

Cited by in corpus (4)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• An inductive proof of Straub's q-analogue of Ljunggren's congruence
• Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas
• Notes on q-analogues of a binomial congruence of Glaisher