Some q-supercongruences from Rahman's summation formula

TL;DR

This paper introduces new $q$-supercongruences inspired by Rahman's summation formula, leading to a Ramanujan-type supercongruence and proposing conjectures on related supercongruences and $q$-supercongruences.

Contribution

It provides novel $q$-supercongruences based on Rahman's quadratic summation formula and derives a new Ramanujan-type supercongruence by taking the limit as $q$ approaches 1.

Findings

Derived new $q$-supercongruences inspired by Rahman's summation formula.

Obtained a Ramanujan-type supercongruence matching Van Hamme's (G.2) for specific primes.

Formulated conjectures on supercongruences and $q$-supercongruences.

Abstract

Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $q\to 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hamme's (G.2) supercongruence for $p\equiv 1 \pmod 4$. We also formulate some related challenging conjectures on supercongruences and $q$-supercongruences.