On divisibility of sums of Apery polynomials

Details On divisibility of sums of Apery polynomials On divisibility of sums of Apery polynomials

2011-08-07

arxiv.org/abs/1108.1546v1

Mathematics

Number Theory

This is a preliminary draft

Scientific paper

For any positive integers $m$ and $\alpha$, we prove that
$$\sum_{k=0}^{n-1}\epsilon^k(2k+1)A_k^{(\alpha)}(x)^m\equiv0\pmod{n}, $$ where
$\epsilon\in\{1,-1\}$ and $$
A_n^{(\alpha)}(x)=\sum_{k=0}^n\binom{n}{k}^{\alpha}\binom{n+k}{k}^{\alpha}x^k.$$

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