q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

Abstract

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.

Incorporated comments from referees. Accepted for publication in Int. J. Number Theory