On congruences involving the generalized Catalan numbers and harmonic numbers
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In this paper, we prove some congruences involving the generalized Catalan numbers and harmonic numbers modulo $p^{2},$ one of which is \begin{align*} \sum\limits_{k=1}^{p-1}k^{2}B_{p,k}B_{p,k-d} \equiv &\ 4\left( -1\right) ^{d}\left\{ \frac{1}{3}d\left( 2d^{2}+1\right) \left( 4pH_{d}-1\right) \right. \\ &\ \left. -p\left( \frac{26}{9}d^{3}+\frac{4}{3}d^{2}+\frac{7}{9}d+\frac{1}{2} \right) \right\}\pmod{p^{2}}, \end{align*} where a prime number $p>3$ and $1\leq d\leq p.$
Abstract
In this paper, we prove some congruences involving the generalized Catalan numbers and harmonic numbers modulo $p^{2},$ one of which is \begin{align*} \sum\limits_{k=1}^{p-1}k^{2}B_{p,k}B_{p,k-d} \equiv &\ 4\left( -1\right) ^{d}\left\{ \frac{1}{3}d\left( 2d^{2}+1\right) \left( 4pH_{d}-1\right) \right. \\ &\ \left. -p\left( \frac{26}{9}d^{3}+\frac{4}{3}d^{2}+\frac{7}{9}d+\frac{1}{2} \right) \right\}\pmod{p^{2}}, \end{align*} where a prime number $p>3$ and $1\leq d\leq p.$
참고문헌 (12)
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