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Register a new userThree supercongruences for Apery numbers or Franel numbers
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Yong Zhang
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The Apery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=sum_{k=0}^{n}{binom{n+k}{2k}}^2{binom{2k}{k}}^2 {rm and } f_n=sum_{k=0}^{n}{binom{n}{k}}^3(n=0, 1, cdots,).$$ In this paper, we prove three supercongruences for Apery numbers or Franel numbers conjectured by Z.-W. Sun. Let $pgeq 5$ be a prime and let $nin mathbb{Z}^{+}$. We show that begin{align} otag frac{1}{n}bigg(sum_{k=0}^{pn-1}(2k+1)A_k-psum_{k=0}^{n-1}(2k+1)A_kbigg)equiv0pmod{p^{4+3 u_p(n)}} end{align} and begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3sum_{k=0}^{n-1}(2k+1)^3A_kbigg)equiv0pmod{p^{6+3 u_p(n)}}, end{align} where $ u_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2sum_{k=0}^{n-1}(3k+2)(-1)^kf_kbigg)equiv0pmod{p^{3}}. end{align}
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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: $$ sum_{k=0}^{(p-1)/2}frac{(frac{1}{2})_k^2}{k!^2} equiv (-1)^{(p-1)/2}+p^2 E_{p-3} pmod{p^3}. $$
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