## 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.

### MSC:

 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.)
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### References:

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