On congruences involving the generalized Catalan numbers and harmonic numbers
Bull. Korean Math. Soc. 2019 Vol. 56, No. 3, 649-658
https://doi.org/10.4134/BKMS.b180454
Published online May 31, 2019
Sibel Koparal, Ne\c{s}e \"{O}m\"{u}r
Kocaeli University; Kocaeli University
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 $1\leq d\leq p.$
Keywords : congruences, harmonic numbers and binomial coefficients
MSC numbers : Primary 11B50, 11A07, 11B65
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