$q$-Analogues of some supercongruences related to Euler numbers

Abstract

Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ \sum_{k=0}^{p-1}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supercongruences by employing the $q$-WZ method. As a conclusion, we provide a $q$-analogue of the following supercongruence of Sun: $$ \sum_{k=0}^{(p-1)/2}\frac{(\frac{1}{2})_k^2}{k!^2} \equiv (-1)^{(p-1)/2}+p^2 E_{p-3} \pmod{p^3}. $$