On congruences involving the generalized Catalan numbers and harmonic numbers
Bull. Korean Math. Soc.
Published online 2019 Jan 25
Sibel Koparal and Neşe Ömür
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{eqnarray*}
\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{eqnarray*}
where a prime number $p>3$ and $1\leq d\leq p$.
Keywords : Congruences, harmonic numbers and binomial coefficients
MSC numbers : 11B50, 11A07, 11B65
Full-Text :

   

Copyright © Korean Mathematical Society. All Rights Reserved.
The Korea Science Technology Center (Rm. 411), 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea
Tel: 82-2-565-0361  | Fax: 82-2-565-0364  | E-mail: paper@kms.or.kr   | Powered by INFOrang.co., Ltd