On congruences involving Apéry numbers

In this paper, we mainly establish a congruence for sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, any prime p &gt; 3 p&gt;3 and positive odd integer alttext="m"> m encoding="application/x-tex">m , prove that there is alttext="p"> encoding="application/x-tex">p -adic alttext="c Subscript m"> c encoding="application/x-tex">c_m only depending on such ∑ k = 0 −<!-- − <mml:mn>1 ( 2 + stretchy="false">) A ≡<!-- ≡ <mml:mo>( ) mod width="0.333em" , encoding="application/x-tex">\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 alttext="upper sigma-summation j StartBinomialOrMatrix Choose EndBinomialOrMatrix squared squared"> j class="MJX-TeXAtom-OPEN"> maxsize="1.2em" minsize="1.2em">( class="MJX-TeXAtom-CLOSE"> minsize="1.2em">) encoding="application/x-tex">A_k=\sum _{j=0}^{k}\binom {k}{j}^2\binom {k+j}{j}^2 the number alttext="left-parenthesis period right-parenthesis"> . encoding="application/x-tex">(\frac {.}{p}) Legendre symbol.