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New congruences with the generalized Catalan numbers and harmonic numbers

Laid Elkhiri|Sibel Koparal|Nese Omur
대한수학회 (The Korean Mathematical Society)|2021.09|대한수학회보 (Bulletin of the KMS)|제58권 제5호|pp. 1079-1095|DOI 10.4134/BKMS.b200359

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참고문헌

키워드

Congruencesharmonic numbers and binomial coefficients
주제분류수학

초록

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.

Abstract

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.

참고문헌 (19)

  1. N. H. Abel, "Untersuchungen ¨uber die Reihe 1+ m 1x+m(m−1)1.2x 2+m(m−1)(m−2)1.2.3x 3+· · ·", Journal für die reine und angewandte Mathematik (Crelles Journal), 1, pp.311~339 (1826), Walter de Gruyter GmbH[DOI]
  2. Horst Alzer, "Series representations for some mathematical constants", Journal of Mathematical Analysis and Applications, 320(1), pp.145~162 (2006), Elsevier BV[DOI]
  3. Xiaojing Chen, "Moments on Catalan numbers", Journal of Mathematical Analysis and Applications, 349(2), pp.311~316 (2009), Elsevier BV[DOI]
  4. Junesang Choi, "Some summation formulas involving harmonic numbers and generalized harmonic numbers", Mathematical and Computer Modelling, 54(9-10), pp.2220~2234 (2011), Elsevier BV[DOI]
  5. Emeric Deutsch, "A survey of the Fine numbers", Discrete Mathematics, 241(1-3), pp.241~265 (2001), Elsevier BV[DOI]
  6. D. H. Greene, "Mathematics for the analysis of algorithms" (1990), Birkh¨auser Boston, Inc
  7. Victor J.W. Guo, "Factors of binomial sums from the Catalan triangle", Journal of Number Theory, 130(1), pp.172~186 (2010), Elsevier BV[DOI]
  8. J.M. Gutiérrez, "New identities in the Catalan triangle", Journal of Mathematical Analysis and Applications, 341(1), pp.52~61 (2008), Elsevier BV[DOI]
  9. Peter Hilton, "Catalan Numbers, Their Generalization, and Their Uses", The Mathematical Intelligencer, 13(2), pp.64~75 (1991), Springer Science and Business Media LLC[DOI]
  10. Sibel Koparal, "On congruences involving the generalized Catalan numbers and harmonic numbers", 대한수학회보, 56(3), pp.649~658 (2019), 대한수학회[DOI]
  11. Emma Lehmer, "On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson", The Annals of Mathematics, 39(2), pp.350~360 (1938), JSTOR[DOI]
  12. P. J. Miana, "Computer proofs of new identities in the Catalan triangle", Proceedings of the “Segundas Jornadas de Teor´ıa de N´umeros”, pp.203~208 (2008), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana
  13. N. Om¨ur, "Some congruences involving numbers Bp,k−d", Util. Math, 95, pp.307~317 (2014)
  14. L.W. Shapiro, "A Catalan triangle", Discrete Mathematics, 14(1), pp.83~90 (1976), Elsevier BV[DOI]
  15. Jurgen Spieb, "Some Identities Involving Harmonic Numbers", Mathematics of Computation, 55(192), pp.839~863 (1990), JSTOR[DOI]
  16. Zhi-Hong Sun, "Congruences concerning Bernoulli numbers and Bernoulli polynomials", Discrete Applied Mathematics, 105(1-3), pp.193~223 (2000), Elsevier BV[DOI]
  17. Z. -W. Sun, "On harmonic numbers and Lucas sequences", Publ. Math. Debrecen, 80(1-2), pp.25~41 (2012)
  18. Zhi-Wei Sun, "Arithmetic theory of harmonic numbers", Proceedings of the American Mathematical Society, 140(2), pp.415~428 (2012), American Mathematical Society (AMS)[DOI]
  19. J. Wolstenholme, "On certain properties of prime numbers", Quart. J. Math, 5, pp.35~39 (1862)

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