Bull. Korean Math. Soc. 2021; 58(5): 1079-1095
Online first article August 26, 2021 Printed September 30, 2021
https://doi.org/10.4134/BKMS.b200359
Copyright © The Korean Mathematical Society.
Laid Elkhiri, Sibel Koparal, Ne\c{s}e \"{O}m\"{u}r
Tiaret University; Kocaeli University; Kocaeli University
In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo $p^{2}.$ One of our results is as follows: for prime number $p>3,$ \begin{align*} &\sum\limits_{k=(p+1)/2}^{p-1}k^{2}B_{p,k}B_{p,k-(p-1)/2}H_{k}\equiv \left( -1\right) ^{(p-1)/2}\left( -\frac{521}{36}p-\frac{1}{p}-\frac{41}{12}\right. \\ &\text{ \ \ \ \ \ \ \ }\left. +pH_{3(p-1)/2}^{2}-10pq_{p}^{2}(2)+4\left( \frac{10}{3}p+1\right) q_{p}(2)\right) \pmod{p^{2}}, \end{align*} where $q_{p}(2)$ is Fermat quotient.
Keywords: Congruences, harmonic numbers and binomial coefficients
MSC numbers: Primary 11B50, 11A07, 11B65
2019; 56(3): 649-658
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