Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON CONGRUENCES INVOLVING THE GENERALIZED CATALAN NUMBERS AND HARMONIC NUMBERS

  • Koparal, Sibel (Department of Mathematics Kocaeli University) ;
  • Omur, Nese (Department of Mathematics Kocaeli University)
  • Received : 2018.05.10
  • Accepted : 2018.11.07
  • Published : 2019.05.31
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Abstract

In this paper, we prove some congruences involving the generalized Catalan numbers and harmonic numbers modulo $p^2$, one of which is $$\sum\limits_{k=1}^{p-1}k^2B_{p,k}B_{p,k-d}{\equiv}4(-1)^d\{{\frac{1}{3}}d(2d^2+1)(4pH_d-1)-p\({\frac{26}{9}}d^3+{\frac{4}{3}}d^2+{\frac{7}{9}}d+{\frac{1}{2}}\)\}\;(mod\;p^2)$$, where a prime number p > 3 and $1{\leq}d{\leq}p$.

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References

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