Some q-supercongruences on double and triple sums

TL;DR

This paper explores new q-supercongruences related to double and triple sums, utilizing advanced methods like creative microscoping and hypergeometric series transformations to establish these congruences.

Contribution

It introduces novel techniques for deriving q-supercongruences on multiple sums, extending existing methods with new lemmas and generalizations.

Findings

Established q-supercongruences for double sums

Derived q-supercongruences for triple sums

Connected supercongruences to hypergeometric series transformations

Abstract

In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the $q$-supercongruences of the square and cube of truncated basic hypergeometric series.Our main tools still involve the `creative microscoping' method introduced by Guo and Zudilin, a lemma designed by Guo and Li and the Chinese remainder theorem.