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

TL;DR

This paper proves a supercongruence conjecture using q-supercongruences, introduces new cyclic sieving phenomena involving q-Lucas numbers, and provides a general criterion based on derivatives at roots of unity.

Contribution

It establishes a supercongruence conjecture through q-supercongruences and uncovers new cyclic sieving phenomena with q-Lucas numbers using a novel criterion.

Findings

Proved a supercongruence conjecture by Almkvist and Zudilin.

Established new q-supercongruences for binomial coefficients and Apéry numbers.

Discovered new cyclic sieving phenomena involving q-Lucas numbers.

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 Ap\'{e}ry 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.