Congruences for Apéry numbers $β_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

Hui-Qin Cao; Yuri Matiyasevich; Zhi-Wei Sun

doi:10.1142/S1793042120500505

paper

Congruences for Apéry numbers $β_{n} = \sum_{k = 0}^{n} (k n)^{2} (k n + k)$

arXiv:1812.10351 · doi:10.1142/S1793042120500505

Abstract

In this paper we establish some congruences involving the Apéry numbers $β_{n} = \sum_{k = 0}^{n} (k n)^{2} (k n + k)$ $(n = 0, 1, 2, \dots)$ . For example, we show that $k = 0 \sum n - 1 (11 k^{2} + 13 k + 4) β_{k} \equiv 0 (mod 2 n^{2})$ for any positive integer $n$ , and $k = 0 \sum p - 1 (11 k^{2} + 13 k + 4) β_{k} \equiv 4 p^{2} + 4 p^{7} B_{p - 5} (mod p^{8})$ for any prime $p > 3$ , where $B_{p - 5}$ is the $(p - 5)$ th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Apery numbers, $β_{n}$ and $A_{n} = \sum_{k = 0}^{n} (k n)^{2} (k n + k)^{2}$ .

arXiv comment23 pages, final published version

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Abstract

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Congruences for Apéry numbers $β_{n} = \sum_{k = 0}^{n} (k n)^{2} (k n + k)$