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Convolution identities involving the central binomial coefficients and Catalan numbers | ||
Transactions on Combinatorics | ||
دوره 10، شماره 4، اسفند 2021، صفحه 225-238 اصل مقاله (437 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/toc.2021.127505.1821 | ||
نویسندگان | ||
Necdet Batır؛ Hakan Kucuk؛ Sezer Sorgun* | ||
Department of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, Turkey | ||
چکیده | ||
We generalize some convolution identities due to Witula and Qi et al. involving the central binomial coefficients and Catalan numbers. Our formula allows us to establish many new identities involving these important quantities, and recovers some known identities in the literature. Also, we give new proofs of Shapiro's Catalan convolution and a famous identity of Haj'{o}s. | ||
کلیدواژهها | ||
Convolution Identity؛ Combinatorial Identity؛ Central Binomial Coefficient؛ Catalan Number؛ Harmonic Number | ||
مراجع | ||
[1] G. Alvarez, J. E. Bergner and R. Lopez, Action graphs and Catalan numbers, J. Integer Seq.,18 2015 pp. 7.
[2] H. Alzer and G. V. Nagy, Some identities involving central binomial coefficients and Catalan numbers, Integers, 20 2020 pp. 17. [3] G. E. Andrews, On Shapiro’s Catalan convolution, Adv. Appl. Math., 46 (2011) 15–24.
[4] R. Apéry, Irrationalité de ζ(2) et ζ(3), Asterisqué, No. 61 (1979) 11–13.
[5] V. De Angelis, Pairings and signed permutations, Amer. Math. Monthly, 113 (2006) 642–644.
[6] K. Ball, MA241 Combinatorics, https://warwick.ac.uk/fac/sci/.
[7] J. M. Borwein and P. B. Borwein, Pi and the AGM: A study in Analytic Number Theory and Computational Complex- ity, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1987. [8] K. N. Boyadzhiev, Series with central binomial coefficients, Catalan numbers, and harmonic numbers, J. Integer Seq., 15 2012 pp. 11. [9] J. M. Campbell, New series involving harmonic numbers and squared central binomial coefficients, Rocky Mountain J. Math., 49 (2019) 2513–2544. [10] G. Chang and C. Xu, Generalization and probabilistic proof of a combinatorial identity, Amer. Math. Monthly, 118 (2017) 175–177. [11] H. Chen, Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers, J. Integer Seq., 19 (2016) pp. 11. [12] R. Duarte and A. G. de Oliveira, A short proof of a famous combinatorial identity, (2013), http://arxiv.org/abs/ 1307.6693. [13] R. Duarte and A. G. Oliveira, A famous identity of Hajós in terms of sets, J. Integer Seq., 17 (2014) pp. 10.
[14] N. Elezović, Asymptotic expansions of central binomial coefficients and Catalan numbers, J. Integer Seq., 17 (2014) pp. 14. [15] P. Hajnal and G. V. Nagy, A bijective proof of Shapiro’s Catalan convolution, Electronic J. Combin., 21 (2014) pp. 10. [16] T. Kosby, Catalan numbers with applications, Oxford Univ. Press, Oxford, 2009.
[17] W.-H. Li, F. Qi, O. Kouba and I. Kaddoura, A further generalization of the Catalan number and its explicit formula and integral representation, preprint, (2020), https://doi.org/10.31219/osf.io/zf9xu. [18] T. Mansour, Y. Sun, Identities involving Narayana polynomials and Catalan numbers, Discrete Math., 309 (2009) 4079–4088. [19] G. V. Nagy, A combinatorial proof of Shapiro’s Catalan convolution, Adv. Appl. Math., 49 (2012) 391–396.
[20] M. Petkovšek, H. S. Wilf and D. Zeilberger, A=B, A. K. Peters, Ltd., Wellesley, Mass., 1996.
[21] F.Qi, C.-P. Chen and D. Lim, Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function, Results in Nonlinear Analysis, 4 (2021) 57-64. [22] F. Qi, Some properties of the Catalan numbers, Ars Combinatoria, in press, https://www.researchgate.net/ publication/328891537. [23] F. Qi, W.-H. Li, J. Cao, D.-W. Niu,and J.-L. Zhao, An analytic generalization of the Catalan numbers and its integral representation, preprint (2020), https:arxiv.org/abs/2005.13515v1. [24] R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers, 6 (2006) pp. 18.
[25] H. M. Srivastava and J. Choi, Zeta and q–zeta Functions and Associated Series and Integrals, Elsevier, 2012.
[26] R. P. Stanley, Catalan Addendum, http://math.mit.edu/~rstan/ec/catadd.pdf.
[27] R. P. Stanley, Catalan Numbers, Cambridge Univ. Press, Cambridge, 2015.
[28] M. Sved, Counting and recounting, Math. intelligencer, 4 (1983) 21–26.
[29] M. Sved, Counting and recounting: The aftermath, Math. intelligencer, 6 (1984) 44–46.
[30] H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3 (1990) 147–158. [31] R. Witula, E. Hetmaniok, D. Slota and N. Gawrońska, Convolution identities for central binomial numbers, Int. J. Pure Appl. Math., 85 (2013) 171–178. | ||
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