We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ for primes $p \geq 5$. 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.

Source : oai:HAL:hal-01215062v1

Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

Section: Proceedings

Published on: January 1, 2011

Submitted on: January 31, 2017

Keywords: $q$-analogs,binomial coefficients,binomial congruence,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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