$q$-Supercongruences for multidimensional series modulo the sixth power of a cyclotomic polynomial

TL;DR

This paper establishes new $q$-supercongruences for multidimensional series modulo the sixth power of a cyclotomic polynomial, extending supercongruence theory with novel methods and applications.

Contribution

It introduces a new approach combining El Bachraoui's lemma, microscoping, and a Chinese remainder theorem variant to prove multidimensional $q$-supercongruences.

Findings

Proved $q$-supercongruences for double and triple series.

Derived supercongruences related to Van Hamme's (D.2) supercongruence.

Extended supercongruence results to higher powers of cyclotomic polynomials.

Abstract

With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series modulo the sixth power of a cyclotomic polynomial. As conclusions, two corresponding supercongruences for double and triple series, which are associated with the (D.2) supercongruence of Van Hamme, are given.