Supercongruences involving Apéry-like numbers and binomial coefficients
Supercongruences involving Apéry-like numbers and binomial coefficients
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DOI：
10.3934/math.2022153
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发表时间：
2021
期刊：
影响因子：
2.2
通讯作者：
Zhi-Hong Sun
中科院分区：
文献类型：
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作者：
Zhi-Hong Sun
设$S_n$是由$S_n=\sum_{k=0}^n\Binom nk\Binom{2k}k\Binom{2n-2k}{n-k}$给出的类APéry序列。证明了对于任意奇素数$p$，$\sum_{n=1}^{p-1}\frac{NS_n}{8^n}{\equv}(1-(-1)^{\frac{p-1}2})p^2\(\Text{mod}\{p^3})$.设$q_n$是由$q_n=\sum_{k=0}^n\Binom nk(-8)^{n-k}\sum_{r=0}^k\Binom KR^3$给出的类APéry数列。我们建立了许多关于$Q_n$的同余。对于奇素数$p$，我们还推导出$\sum_{k=0}^{p-1}\Binom{2k}k^3\frc 1{^k}\(\Text{mod}\{p^3})$，$SUM_{k=0}^{p-1}\Binom{2k}k^3\FRAC 1{^k(k+1)^2}\(\Text{mod}\{p^2})$和$\sum_{k=0}^{p-1}\Binom{2k}k^3\FRAC 1{^k(2k-1)}\(\Text{mod}\p)$，并对含有二项式系数和类APéry数的同余提出了许多猜想。
Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.