Congruences involving Franel and Catalan-Larcombe-French numbers
Congruences involving Franel and Catalan-Larcombe-French numbers
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发表时间：
2015-02
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通讯作者：
Zhi-Hong Sun
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作者：
Zhi-Hong Sun
设$\{f_n\}$为$f_n=\sum_{k=0}^n\binom nk^3$给出的弗兰内尔数，设$p>5$为素数。本文主要确定$m=5,-16,16,32,-49,50,96$为$\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$。让$S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$。我们还为$m=7,16,25,32,64,160,800,1600, 156832$确定了$\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$。
Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let $S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$. We also determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$ for $m=7,16,25,32,64,160,800,1600, 156832$.