On congruences involving Apéry numbers

Abstract

In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime p > 3 p>3 and positive odd integer m m , we prove that there is a p p -adic integer c m c_m only depending on m m such that ∑ k = 0 p − 1 ( 2 k + 1 ) m ( − 1 ) k A k ≡ c m p ( p 3 ) ( mod p 3 ) , \begin{equation*} \sum _{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left (\frac {p}{3}\right )\pmod {p^3}, \end{equation*} where A k = ∑ j = 0 k ( k j ) 2 ( k + j j ) 2 A_k=\sum _{j=0}^{k}\binom {k}{j}^2\binom {k+j}{j}^2 is the Apéry number and ( . p ) (\frac {.}{p}) is the Legendre symbol.