## A $$q$$-analog of Ljunggren’s binomial congruence.(English. French summary)Zbl 1355.05053

Proceedings of the 23rd international conference on formal power series and algebraic combinatorics, FPSAC 2011, Reykjavik, Iceland, June 13–17, 2011. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 897-902 (2011).
Summary: We prove a $$q$$-analog of a classical binomial congruence due to Ljunggren which states that ${ap \choose bp} \equiv {a \choose b}$ modulo $$p^3$$ for primes $$p \geqslant 5$$. 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 W. E. Clark [Int. J. Math. Math. Sci. 18, No. 1, 197–200 (1995; Zbl 0816.05009)].
For the entire collection see [Zbl 1239.05002].

### MSC:

 05A30 $$q$$-calculus and related topics 05A10 Factorials, binomial coefficients, combinatorial functions

### Keywords:

$$q$$-analogs; binomial coefficients; binomial congruence

Zbl 0816.05009
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