Supercongruences for polynomial analogs of the Apéry numbers

Abstract

We consider a family of polynomial analogs of the Apéry numbers, which includes q-analogs of Krattenthaler-Rivoal-Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Apéry 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. *

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

“…Finally, by taking the limit a → 1, we obtain the original q-supercongruence of interest. We learned that this creative microscoping method has already caught the interests of Guillera [14] and Straub [50].…”

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

“…So one possible way to prove that a sequence satisfies the Gauss congruences is to find a q-analogue of it that satisfies the q-Gauss congruences. As demonstrated in recent works of Guo and Zudilin [GZ18] and Straub [Str18], the approach of establishing congruences via q-congruences is fruitful because of additional techniques available in the q-setting. In this work we make heavy use of the derivative and its properties.…”

A New Family of q-Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial

“…During the past few years, q-analogues of congruences and supercongruences have been investigated by many authors (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16]18,21,22,24,25,27]). For instance, using a method similar to that used in [26], the first author and Wang [12,Theorem 1.2] gave a q-analogue of (1.2): for odd n,…”