Some variations of a "divergent" Ramanujan-type $q$-supercongruence

Abstract

Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$\sum _{k=0}^{p-1}(3k-1)\frac{(\frac{1}{2}{)}_k(-\frac{1}{2}{)}_k^2}{k{!}^3}{4}^k\equiv p-2p^3(\mathrm{mod}{p}^4),$$ where $(a{)}_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.

10 pages

References in corpus (3)

• Congruences on sums of $q$-binomial coefficients
• Symbolic summation methods and hypergeometric supercongruences
• Some $q$-congruences arising from certain identities