A family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial

TL;DR

This paper establishes a new family of $q$-hypergeometric congruences modulo the fourth power of cyclotomic polynomials, using advanced transformation and summation techniques, and confirms two prior conjectures.

Contribution

It introduces a novel two-parameter family of $q$-hypergeometric congruences and proves two conjectures using sophisticated hypergeometric series identities.

Findings

Proved a new family of $q$-congruences modulo cyclotomic polynomial powers.

Validated two previously conjectured $q$-congruences.

Utilized advanced hypergeometric transformation and summation formulas.

Abstract

We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.