Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas

TL;DR

This paper develops q-analogues of classical supercongruences related to binomial coefficients, extending previous results and employing advanced hypergeometric series techniques to prove new congruences modulo cyclotomic polynomials.

Contribution

The paper introduces novel q-analogues of supercongruences of Beukers, Van Hamme, and Rodriguez-Villegas, using hypergeometric series methods and establishing new q-congruences.

Findings

Proved q-analogues of Van Hamme's supercongruence for primes p ≡ 3 mod 4.

Established q-analogues of Rodriguez-Villegas' supercongruences for specific prime residue classes.

Utilized a new q-Clausen-type summation formula in the proofs.

Abstract

For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\frac{1}{64^k} \equiv 0 \pmod{p^2} \quad\text{for}\quad p\equiv 3\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^2 } &\equiv 0 \pmod{[p]^2}\quad\text{for}\quad p\equiv 2\pmod{3}, \end{align*} where $[p]=1+q+\cdots+q^{p-1}$, $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, and ${n\brack k}_q$ denotes the q-binomial coefficient. Actually, our results give q-analogues of Z.-H. Sun's and Z.-W. Sun's generalizations of the above Beukers-like supercongruences. Our proof uses the theory of basic hypergeometric series including a new q-Clausen-type summation formula.