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$q$-Analogues of some supercongruences related to Euler numbers (2010.13526v1)

Published 20 Oct 2020 in math.NT and math.CO

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 $$ \sum{(p-1)/2}{k=0}(-1)k(4k+1)\frac{(\frac{1}{2})_k3}{k!3} \equiv p(-1){(p-1)/2}+p3E_{p-3} \pmod{p4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ \sum_{k=0}{p-1}(-1)k (3k+1)\frac{(\frac{1}{2})k3}{k!3} 2{3k} \equiv p(-1){(p-1)/2}+p3E{p-3} \pmod{p4}, $$ 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: $$ \sum_{k=0}{(p-1)/2}\frac{(\frac{1}{2})_k2}{k!2} \equiv (-1){(p-1)/2}+p2 E_{p-3} \pmod{p3}. $$

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