Some q-analogues of supercongruences of Rodriguez-Villegas

Abstract

We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv (-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \varepsilon=\pm1. Several related conjectures are also proposed.

14 pages, to appear in J. Number Theory

References in corpus (2)

• Congruences concerning Legendre polynomials
• A q-analogue of Wolstenholme's harmonic series congruence

Cited by in corpus (13)

• A $q$-microscope for supercongruences
• Dwork-type supercongruences through a creative $q$-microscope
• Some new $q$-congruences for truncated basic hypergeometric series
• Supercongruences on some binomial sums involving Lucas sequences
• On two conjectural supercongruences of Z.-W. Sun
• Congruences for truncated hypergeometric series ${}_2{F}_1$
• New congruences involving harmonic numbers
• The Rodriguez-Villegas type congruences for truncated q-hypergeometric functions
• Proof of a conjectural supercongruence
• Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas
• Factors of some truncated basic hypergeometric series
• Factors of alternating sums of powers of $q$-Narayana numbers
• Some symmetric $q$-congruences