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Articles

Combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbers

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Pages 222-243 | Received 22 Jul 2021, Accepted 26 Jan 2022, Published online: 19 Feb 2022

References

  • N. Batır, Finite binomial sum identities with harmonic numbers, J. Integer Seq. 24 (2021), p. 22. Article 21.4.3.
  • N. Batır, Combinatorial identities involving harmonic numbers, Integers 20 (2020), p. 18 . #A25.
  • N. Batır, On some combinatorial identities and harmonic sums, Int. J. Number Theor. 13 (2017), pp. 1695–1709.
  • N. Batır, On the series ∑k=1∞(3kk)−1k−nxk, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), pp. 371–381.
  • N. Batır and K. - W. Chen, Finite Hurwitz-Lerch functions, Filomat 33 (2019), pp. 101–109.
  • N. Batır and A. Sofo, A unified treatment of certain classes of combinatorial identities with harmonic numbers, J. Integer Seq. 24 (2021), p. 3. Article 21.3.2.
  • N. Batır and A. Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comput. 220 (2013), pp. 331–338.
  • H. Belbachir and M. Rahmani, Alternating sums of the reciprocals of binomial coefficients, J. Integer Seq. 15 (2012), p. 3. Article 12.2.8.
  • H. Belbachir, M. Rahmani, and B. Sury, Sums involving moments of reciprocals of binomial coefficients, J. Integer Seq. 14 (2011), pp. 1–16, Article 11.6.6.
  • K. N. Boyadzhiev, Series with central binomial coefficients, Catalan numbers, and harmonic numbers, J. Integer Seq. 15 (2012), pp. 1–11, Article 12.1.7 .
  • J.M. Campbell, Identities for finite sums involving central binomial coefficients and harmonic-type numbers, preprint (2021).
  • H. Chen, Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers, J. Integer Seq. 19 (2016), p. 16. Article 16.1.5.
  • W. Chu and L. De Donno, Hypergeometric series and harmonic number identities, Adv. Appl. Math.34 (2005), pp. 123–137.
  • P. Duren, Invitation to Classical Analysis, Amer. Math. Soc., 2012.
  • R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, New York, 1994.
  • H.T. Jin and D.K. Du, Abel's lemma and identities on harmonic numbers, Integers. 15 (2015), p. #A22.
  • M. Kauers and C. Schneider, Application of unspecified sequences in symbolic summation, in Proc. ISSAC'06, J. Dumas, ed., ACM Press, 2006, pp. 177–183.
  • M. Kauers and C. Schneider, Indefinite summation with unspecified summands, Discrete Math. 306(17) (2006), pp. 2073–2083.
  • T. Mansour, Combinatorial identities and inverse binomial coefficients, Adv. Appl. Math. 28 (2002), pp. 196–202. 1391–1399.
  • P. Paule and C. Schneider, Towards a symbolic summation theory for unspecified sequences, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, J. Blümlein, P. Paule, and C. Schneider, eds., Texts and Monographs in Symbolic Computation, Springer, 2019, pp. 351–390. Available at arXiv: 1809.06578 [cs.SC], doi:https://doi.org/10.1007/978-3-030–4480-0_15
  • J. Pla, The sum of inverses of binomial coefficients revisited, Fib. Quart. 35 (1997), pp. 342–345.
  • M.A. Rockett, Sums of the inverses of binomial coefficients, Fib. Quart. 19 (1981), pp. 433–445.
  • C. Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin. 56 (2007), pp. 1–36. Article B56b.
  • R. Sprugnoli, Alternating weighted sums of inverse of binomial coefficients, J. Integer Seq. 15 (2012), p. 3. Article 12.6.3.
  • R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers 6 (2006), pp. 16.
  • H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012.
  • B. Sury, Sum of the reciprocals of the binomial coefficients, European J. Combin. 14 (1993), pp. 351–353.
  • B. Sury, T. Wang, and F. Z. Zhao, Some identities involving reciprocals of binomial coefficients, J. Integer Seq. 7 (2004), p. 3. Article 04.2.8.
  • T. Trif, Combinatorial sums and series involving the inverses of binomial coefficients, Fib. Quart. 38 (2000), pp. 79–84.
  • R. Witula and D. Slota, Finite sums connected with the inverses of central binomial numbers and Catalan numbers, Asian-Eur. J. Math. 1 (2008), pp. 439–448.
  • R. Witula, D. Slota, J. Watlak, A. Chmielowska, and M. Rózański, Matrix methods in evaluation of integrals, J. Appl. Math. Comput. Mech. 19 (2020), pp. 103–112.
  • J.H. Yang and F.Z. Zhao, Certain sums involving inverses of binomial coefficients and some integrals, J. Integer Seq. 10 (2007), p. 3. Article 07.8.7.

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