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.

New q-Analogues of Van Hamme’s (E.2) Supercongruence and of a Supercongruence by Swisher

“…During the past few years, q-analogues of supercongruences have been investigated by many authors (see, for example, [3][4][5][6][7][8][9][10][12][13][14][15]19,20,22,[25][26][27]29]). In particular, the first author [3,4] gave q-analogues of (1.1) and (1.2) as follows: for any odd integer n,…”

“…Note that the supercongruence (1.5) does not hold modulo [n]Φ n (q) 2 in general, even for n ≡ 1 (mod 6). We take this opportunity to point out that Theorems 1 and 2 in [8] only hold modulo Φ n (q) 3 and Φ n (q) 2 , respectively, but do not hold modulo [n], since Lemma 3 in [8] is not true (it only holds for even integers d).…”