Some \(q\)-supercongruences for truncated basic hypergeometric series. (English) Zbl 1338.11024

Summary: For any odd prime \(p\) we obtain \(q\)-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as \[ \begin{aligned} \sum_{k=0}^{{(p-1)}/{2}} \bigg[{2k\atop k}\bigg]_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0 \pmod{[p]^2} \;\text{for}\;p\equiv 3 \pmod 4, \\ \sum_{k=0}^{{(p-1)}/{2}}\bigg[{2k\atop k}\bigg]_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k}}{ (q^{6};q^{6})_k^2} &\equiv 0 \pmod{[p]^2}\;\text{for}\;p\equiv 2 \pmod{3}, \end{aligned} \] where \([p]=1+q+\cdots+q^{p-1}\) and \((a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})\). We also prove \(q\)-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial \(q\)-binomial identities including a new \(q\)-Clausen type summation formula.


11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
05A30 \(q\)-calculus and related topics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI


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