Supercongruences for polynomial analogs of the Ap\'ery numbers

TL;DR

This paper extends known supercongruences from Apéry numbers to polynomial analogs, including q-analogs, using classical binomial congruences, and suggests broader applicability to other supercongruence results.

Contribution

It introduces a method to prove supercongruences for polynomial analogs of Apéry numbers, generalizing previous results and classical binomial congruences.

Findings

Supercongruences for polynomial analogs of Apéry numbers are established.

The approach generalizes to other supercongruence results.

Classical binomial congruences underpin the proofs.

Abstract

We consider a family of polynomial analogs of the Ap\'ery numbers, which includes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Ap\'ery numbers generalize to these polynomials. Our proof relies on polynomial analogs of classical binomial congruences of Wolstenholme and Ljunggren. We further indicate that this approach generalizes to other supercongruence results.