$q$-Supercongruences from squares of basic hypergeometric series

TL;DR

This paper establishes new $q$-supercongruences involving squares of basic hypergeometric series, primarily modulo high powers of cyclotomic polynomials, using advanced proof techniques and proposing conjectures for future research.

Contribution

It introduces novel $q$-supercongruences for squares of basic hypergeometric series, expanding the understanding of their congruence properties.

Findings

Most supercongruences are modulo the cube of a cyclotomic polynomial.

Two supercongruences are modulo the fourth power of a cyclotomic polynomial.

The paper proposes several conjectures for further exploration.

Abstract

We give some new $q$-supercongruences on truncated forms of squares of basic hypergeometric series. Most of them are modulo the cube of a cyclotomic polynomial, and two of them are modulo the fourth power of a cyclotomic polynomial. The main ingredients of our proofs are the creative microscoping method, a lemma of El Bachraoui, and the Chinese remainder theorem for coprime polynomials. We also propose several related conjectures for further study.