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

References in corpus (3)

• Supercongruences for sporadic sequences
• The generalized Borwein conjecture. II. Refined q-trinomial coefficients
• A supercongruence for generalized Domb numbers

Cited by in corpus (7)

• Dwork-type supercongruences through a creative $q$-microscope
• Congruences for $q$-binomial coefficients
• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• A common $q$-analogue of two supercongruences
• Dold sequences, periodic points, and dynamics
• New representations for all sporadic Apéry-like sequences, with applications to congruences
• On sporadic sequences