Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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NEW CONGRUENCES WITH THE GENERALIZED CATALAN NUMBERS AND HARMONIC NUMBERS

  • Elkhiri, Laid (Department of Mathematics Recits Laboratory USTHB Bab Ezzouar Tiaret University) ;
  • Koparal, Sibel (Department of Mathematics Kocaeli University) ;
  • Omur, Nese (Department of Mathematics Kocaeli University)
  • Received : 2020.04.19
  • Accepted : 2021.06.04
  • Published : 2021.09.30
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Abstract

In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo p2. One of our results is as follows: for prime number p > 3, $${\sum\limits_{k=(p+1)/2}^{p-1}}\;k^2B_{p,k}B_{p,k-(p-1)/2}H_k{\equiv}(-1)^{(p-1)/2}\(-{\frac{521}{36}}p-{\frac{1}{p}}-{\frac{41}{12}}+pH^2_{3(p-1)/2}-10pq^2_p(2)+4\({\frac{10}{3}}p+1\)q_p(2)\)\;(mod\;p^2),$$ where qp(2) is Fermat quotient.

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Acknowledgement

The author would like to thank the referee for carefully reading the paper and valuable comments to improve its presentation.

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