$q$-Supercongruences from Gasper and Rahman's summation formula

TL;DR

This paper proves two $q$-supercongruences related to hypergeometric series, confirming one of He's conjectures and generalizing the other, using advanced methods like microscoping and the Chinese remainder theorem.

Contribution

It introduces new $q$-supercongruences derived from Gasper and Rahman's summation formula, confirming and extending previous conjectures.

Findings

Confirmed He's first conjecture on $q$-supercongruences.

Proved a more general form of He's second conjecture.

Derived supercongruences modulo high powers of cyclotomic polynomials.

Abstract

In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several $q$-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for basic hypergeometric series. As conclusions, He's first conjecture is confirmed and a more general form of He's second conjecture is proved.