Three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial

Abstract

In this paper, three parametric q-supercongruences for truncated very-well-poised basic hypergeometric series are proved, one of them modulo the square, the other two modulo the cube of a cyclotomic polynomial. The main ingredients of proof include a basic hypergeometric summation by George Gasper, the method of creative microscoping (a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

Keywords: Basic hypergeometric series; Cyclotomic polynomial; Gasper’s summation; Supercongruences; q-congruences.