Some q -supercongruences from a~very-well-poised 6 φ 5 summation

Le résumé fourni par la source

Abstract By making use of a very-well-poised ϕ 5 6 {{}_{6}\phi_{5}} summation, the creative microscoping method, and the Chinese remainder theorem for coprime polynomials, we establish some new q -supercongruences, including some Dwork-type q -supercongruences. As a consequence, we obtain the following result: for any prime p ≡ 1 ( mod 4 ) {p\equiv 1\pmod{4}} and integer r ⩾ 1 {r\geqslant 1} , ∑ k = 0 p r - 1 2 ( 8 ⁢ k + 1 ) ⁢ ( 1 4 ) k 3 ⁢ ( 1 2 ) k k ! 3 ⁢ ( 3 4 ) k ≡ p r ( mod p r + 3 ) , \sum_{k=0}^{\frac{p^{r}-1}{2}}(8k+1)\frac{(\frac{1}{4})_{k}^{3}(\frac{1}{2})_{% k}}{k!^{3}(\frac{3}{4})_{k}}\equiv p^{r}\pmod{p^{r+3}},

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