Long Li: Two $q$-supercongruences from Jackson's ${}_6\phi_5$ summation, 169-175

Abstract:

Employing Jackson's ${}_6\phi_5$ summation and the `creative microscoping' method introduced by Guo and Zudilin, we prove two $q$-supercongruences modulo the cube of a cyclotomic polynomial. As conclusions, we obtain two supercongruences, one of which can be stated as follows: for any prime $p\equiv 1 \pmod 4$ and $0\leqslant s\leqslant (p-1)/4$, modulo $p^3$,

$$\sum_{k=s}^{(3p-3)/4} (8k+3)\frac{(\frac{3}{4})_{k-s}(\frac{3}{4... ...\frac{3}{2})_{(3p-3)/4-s}} {s!^2(1)_{(3p-3)/4+s}(\frac{3}{4})_{(3p-3)/4-s}},$$

where $(x)_n= x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol.

Key Words: $q$-supercongruences, supercongruences, Jackson's ${}_6\phi_5$ summation, creative microscoping.

2020 Mathematics Subject Classification: Primary 11A07; Secondary 11B65, 33D15.

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