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Research Article

Some combinatorial identities containing central binomial coefficients or Catalan numbers*

, &
Article: 2204233 | Received 25 Nov 2022, Accepted 11 Apr 2023, Published online: 30 Apr 2023

References

  • Alzer H, Nagy GV. Some identities involving central binomial coefficients and Catalan numbers. Integers. 2020;20:A59.
  • Boyadzhiev KN. Series with central binomial coefficients, Catalan numbers, and harmonic numbers. J Integer Seq. 2012;15(1):12.1.7.
  • Campbell JM. New series involving harmonic numbers and squared central binomial coefficients. Rocky Mountain J Math. 2019;49(8):2513–2544. https://doi.org/10.1216/RMJ-2019-49-8-2513.
  • Chen H. Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers. J Integer Seq. 2016;19(1):16.1.5.
  • Garcia-Armas M, Seturaman BA. A note on the Hankel transform of the central binomial coefficients. J Integer Seq. 2008;11:08.5.8.
  • Gu CY, Guo VJW. Proof of two conjectures on supercongruences involving central binomial coefficients. Bull Aust Math Soc. 2020;102(3):360–364. https://doi.org/10.1017/s0004972720000118.
  • Li Y-W, Qi F. A sum of an alternating series involving central binomial numbers and its three proofs. J Korean Soc Math Educ Ser B Pure Appl Math. 2022;29(1):31–35. https://doi.org/10.7468/jksmeb.2022.29.1.31.
  • Mikić J. On certain sums divisible by the central binomial coefficient. J Integer Seq. 2020;23(1):20.1.6.
  • Qi F, Lim D. Integral representations and properties of several finite sums containing central binomial coefficients. ScienceAsia. 2023;49(2):205–211. http://dx.doi.org/10.2306/scienceasia1513-1874.2022.137.
  • Spivey MZ. The art of proving binomial identities. Boca Raton, FL: CRC Press; 2019. (Discrete Mathematics and its Applications).
  • Sprugnoli R. Sums of reciprocals of the central binomial coefficients. Integers. 2006;6:A27.
  • Abramowitz M, Stegun IA. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York, Washington: Dover Publications; 1972 (Applied Mathematics Series; vol. 55).
  • Adams EP, Hippisley RL. Smithsonian mathematical formulae and tables of elliptic functions. Washington, D.C.: Smithsonian Institute; 1922.
  • Berndt BC. Ramanujan's notebooks, part I. New York: Springer-Verlag; 1985.
  • Borwein JM, Bailey DH, Girgensohn R. Experimentation in mathematics: computational paths to discovery. Natick, MA: A K Peters, Ltd.; 2004.
  • Borwein JM, Borwein PB. Pi and the AGM: a study in analytic number theory and computational complexity. New York: A Wiley-Interscience Publication, John Wiley & Sons, Inc.; 1987. (Canadian Mathematical Society Series of Monographs and Advanced Texts).
  • Chen CP. Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions. Integral Transforms Spec Funct. 2012;23(12):865–873. https://doi.org/10.1080/10652469.2011.644851.
  • Davydychev AI, Kalmykov MY. New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams. Nuclear Phys B. 2001;605(1-3):266–318. https://doi.org/10.1016/S0550-3213(01)00095-5.
  • Edwards J. Differential calculus. 2nd ed. London: Macmillan; 1982.
  • Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Amsterdam: Elsevier/Academic Press; 2015. Available Online athttps://doi.org/10.1016/B978-0-12-384933-5.00013-8.
  • Lehmer DH. Interesting series involving the central binomial coefficient. Amer Math Monthly. 1985;92(7):449–457. http://dx.doi.org/10.2307/2322496.
  • Qi F, Guo BN. Integral representations of the Catalan numbers and their applications. Mathematics. 2017;5(3):40. https://doi.org/10.3390/math5030040.
  • Wilf HS. Generatingfunctionology, 3rd ed. Wellesley, MA: A K Peters, Ltd.; 2006.
  • Zhang B, Chen C-P. Sharp Wilker and Huygens type inequalities for trigonometric and inverse trigonometric functions. J Math Inequal. 2020;14(3):673–684. https://doi.org/10.7153/jmi-2020-14-43.
  • Grimaldi RP. Fibonacci and Catalan numbers. Hoboken, NJ: John Wiley & Sons, Inc.; 2012.
  • Koshy T. Catalan numbers with applications. Oxford: Oxford University Press; 2009.
  • Roman S. An introduction to catalan numbers. Cham: Birkhäuser/Springer; 2015. (Compact Textbook in Mathematics).
  • Stanley RP. Catalan numbers. New York: Cambridge University Press; 2015.
  • Bromwich TJI. An introduction to the theory of infinite series. London: Macmillan and Co., Ltd.; 1908.
  • Bradley DM. A class of series acceleration formulae for Catalan's constant. Ramanujan J. 1999;3(2):159–173https://doi.org/10.1023/A:1006945407723.
  • Qi F, Chen CP, Lim D. Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. Results Nonlinear Anal. 2021;4(1):57–64. https://doi.org/10.53006/rna.867047.
  • Guo BN, Lim D, Qi F. Series expansions of powers of arcsine, closed forms for special values of bell polynomials, and series representations of generalized logsine functions. AIMS Math. 2021;6(7):7494–7517. https://doi.org/10.3934/math.2021438.
  • Guo BN, Lim D, Qi F. Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial bell polynomials, and series representation of generalized logsine function. Appl Anal Discrete Math. 2022;16(2):427–466. https://doi.org/10.2298/AADM210401017G.
  • Qi F. Explicit formulas for partial bell polynomials, Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine, and series representations of powers of Pi. Research Square. 2021. https://doi.org/10.21203/rs.3.rs-959177/v3.
  • Qi F. Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial bell polynomials, and series representations for real powers of Pi. Demonstr Math. 2022;55(1):710–736. https://doi.org/10.1515/dema-2022-0157.
  • Qi F, Ward MD. Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function composited by inverse tangent, square root, and exponential functions. arXiv (2022). https://arxiv.org/abs/2110.08576v2.
  • Abel U. Reciprocal Catalan sums: Solution to Problem 11765. Amer Math Monthly. 2016;123(4):405–406. https://doi.org/10.4169/amer.math.monthly.123.4.399.
  • Amdeberhan T, Guan X, Jiu L, et al. A series involving Catalan numbers: proofs and demonstrations. Elem Math. 2016;71(3):109–121. https://doi.org/10.4171/EM/306.
  • Beckwith D, Harbor S. Problems and solutions. Amer Math Monthly. 2014;121(3):267–267. https://doi.org/10.4169/amer.math.monthly.121.03.266.
  • Koshy T, Gao ZG. Convergence of a Catalan series. College Math J. 2012;43(2):141–146. https://doi.org/10.4169/college.math.j.43.2.141.
  • Stewart SM. The inverse versine function and sums containing reciprocal central binomial coefficients and reciprocal Catalan numbers. Int J Math Educ Sci Technol. 2022;53(7):1955–1966. https://doi.org/10.1080/0020739X.2021.1912842.
  • Olver FWJ, Lozier DW, Boisvert RF, et al. NIST handbook of mathematical functions. New York: Cambridge University Press; 2010.