$q$-Supercongruences on triple and quadruple sums

TL;DR

This paper establishes new $q$-supercongruences involving triple and quadruple sums of basic hypergeometric series, extending classical supercongruences to the $q$-analogue setting with higher power modulus.

Contribution

It introduces novel $q$-supercongruences on multiple sums, including a $q$-analogue of Van Hamme's quadruple sum supercongruence, advancing the understanding of hypergeometric series in modular settings.

Findings

Proves a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial.

Provides a $q$-analogue of Van Hamme's supercongruence (G.2).

Extends supercongruence results to triple and quadruple sums in the $q$-hypergeometric context.

Abstract

Inspired by the recent work of El Bachraoui, we present some new $q$-supercongruences on triple and quadruple sums of basic hypergeometric series. In particular, we give a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial, which is a $q$-analogue of the quadruple sum of Van Hamme's supercongruence (G.2).