Some $q$-supercongruences for multiple basic hypergeometric series

TL;DR

This paper establishes new $q$-supercongruences for multiple basic hypergeometric series using advanced summation, transformation formulas, and the creative microscoping method, extending previous supercongruences related to Van Hamme's conjectures.

Contribution

It generalizes existing $q$-supercongruences for double hypergeometric series and introduces new results for double and triple series related to Van Hamme's supercongruences.

Findings

Generalized Guo and Li's $q$-supercongruences for double series.

Established supercongruences modulo fifth and sixth powers of cyclotomic polynomials.

Presented new supercongruences for double and triple hypergeometric series.

Abstract

In terms of several summation and transformation formulas for basic hypergeometric series, two forms of the Chinese remainder theorem for coprime polynomials, the creative microscoping method introduced by Guo and Zudilin, Guo and Li's lemma, and El Bachraoui's lemma, we establish some $q$-supercongruences for multiple basic hypergeometric series modulo the fifth and sixth powers of a cyclotomic polynomial. In detail, we generalize Guo and Li's two $q$-supercongruences for double basic hypergeometric series, which are related to $q$-analogues of Van Hamme's (C.2) supercongruence and Long's supercongruence, respectively. In addition, we also present two conclusions for double and triple hypergeometric series associated with Van Hamme's (D.2) supercongruence.