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Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function

Yıl 2021, Cilt: 4 Sayı: 1, 57 - 64, 31.03.2021

Feng Qi , Chao-ping Chen , Dongkyu Lım

https://doi.org/10.53006/rna.867047
Cited By: 8

Öz

In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are related to the Catalan numbers in combinatorial number theory.

Anahtar Kelimeler

identity product ratio central binomial coefficient power series expansion arcsine function square cubic generating function Catalan number

Kaynakça

  • [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, New York and Washington, 1972.
  • [2] E. P. Adams and R. L. Hippisley, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922.
  • [3] H. Alzer and G. V. Nagy, Some identities involving central binomial coeficients and Catalan numbers, Integers 20 (2020), Paper No. A59, 17 pages.
  • [4] B. C. Berndt, Ramanujan's Notebooks, Part I, With a foreword by S. Chandrasekhar, Springer-Verlag, New York, 1985; available online at https://doi.org/10.1007/978-1-4612-1088-7.
  • [5] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
  • [6] J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
  • [7] K. N. Boyadzhiev, Series with central binomial coeficients, Catalan numbers, and harmonic numbers. J. Integer Seq. 15 (2012), no. 1, Article 12.1.7, 11 pp.
  • [8] D. M. Bradley, A class of series acceleration formulae for Catalan's constant, Ramanujan J. 3 (1999), no. 2, 159-173; available online at https://doi.org/10.1023/A:1006945407723.
  • [9] T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan and Co., Limited, London, 1908.
  • [10] J. M. Campbell, New series involving harmonic numbers and squared central binomial coeficients, Rocky Mountain J. Math. 49 (2019), no. 8, 2513-2544; available online at https://doi.org/10.1216/RMJ-2019-49-8-2513.
  • [11] H. Chen, Interesting series associated with central binomial coeficients, Catalan numbers and harmonic numbers, J. Integer Seq. 19 (2016), no. 1, Article 16.1.5, 11 pp.
  • [12] C.-P. Chen, Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions, Integral Transforms Spec. Funct. 23 (2012), no. 12, 865-873; available online at https://doi.org/10.1080/10652469.2011.644851.
  • [13] P. P. Dályay, Y. J. Ionin, O. P. Lossers, and J. H. Smith, A product of Catalan numbers, Amer. Math. Monthly 125 (2018), no. 1, 86-87; available online at https://doi.org/10.1080/00029890.2018.1397465.
  • [14] A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B 699 (2004), no. 1-2, 3-64; available online at https://doi.org/10.1016/j.nuclphysb.2004.08.020.
  • [15] A. I. Davydychev and M. Yu. Kalmykov, New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams, Nuclear Phys. B 605 (2001), no. 1-3, 266-318; available online at https://doi.org/10.1016/S0550-3213(01) 00095-5.
  • [16] J. Edwards, Differential Calculus, 2nd ed., Macmillan, London, 1982.
  • [17] M. Garcia-Armas and B. A. Seturaman, A note on the Hankel transform of the central binomial coeficients, J. Integer Seq. 11(2008), Article 08.5.8, 9 pages.
  • [18] I. S. Gradshteyn and I. M. Ryzhik, 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, Elsevier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
  • [19] R. P. Grimaldi, Fibonacci and Catalan Numbers, John Wiley & Sons, Inc., Hoboken, NJ, 2012; available online at https: //doi.org/10.1002/9781118159743.
  • [20] C.-Y. Gu and V. J. W. Guo, Proof of two conjectures on supercongruences involving central binomial coeficients, Bull. Aust. Math. Soc. 102 (2020), no. 3, 360-364; available online at https://doi.org/10.1017/s0004972720000118.
  • [21] B.-N. Guo, D. Lim, and F. Qi, Maclaurin series expansions for powers of inverse (hyperbolic) sine, for powers of inverse (hyperbolic) tangent, and for incomplete gamma functions, with applications to second kind Bell polynomials and generalized logsine function, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.10686v5.
  • [22] M. Yu. Kalmykov and A. Sheplyakov, lsjka C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions, Computer Phys. Commun. 172 (2005), no. 1, 45-59; available online at https://doi.org/10.1016/j. cpc.2005.04.013.
  • [23] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009.
  • [24] D. H. Lehmer, Interesting series involving the central binomial coeficient, Amer. Math. Monthly 92 (1985), no. 7, 449-457; available online at http://dx.doi.org/10.2307/2322496.
  • [25] W.-H. Li, F. Qi, O. Kouba, and I. Kaddoura, A further generalization of the Catalan numbers and its explicit formula and integral representation, OSF Preprints (2020), available online at https://doi.org/10.31219/osf.io/zf9xu.
  • [26] M. Mahmoud and F. Qi, Three identities of the Catalan-Qi numbers, Mathematics 4 (2016), no. 2, Article 35, 7 pages; available online at https://doi.org/10.3390/math4020035.
  • [27] J. Miki¢, On certain sums divisible by the central binomial coeficient, J. Integer Seq. 23 (2020), no. 1, Art. 20.1.6, 22 pages.
  • [28] F. Qi, Some properties of the Catalan numbers, Ars Combin. (2022), in press; available online at https://www. researchgate.net/publication/328891537.
  • [29] F. Qi and P. Cerone, Some properties of the Fuss-Catalan numbers, Mathematics 6 (2018), no. 12, Article 277, 12 pages; available online at https://doi.org/10.3390/math6120277.
  • [30] F. Qi, C.-P. Chen, and D. Lim, Five identities involving the product or ratio of two central binomial coeficients, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.02027v1
  • [31] F. Qi, C.-P. Chen, and D. Lim, Several combinatorial identities derived from series expansions of powers of arcsine, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.02027v2.
  • [32] F. Qi and B.-N. Guo, From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions, J. Math. Anal. Appl. 493 (2021), no. 1, Article 124478, 19 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124478.
  • [33] F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (2017), no. 3, Article 40, 31 pages; available online at https://doi.org/10.3390/math5030040.
  • [34] F. Qi and B.-N. Guo, Sums of infinite power series whose coeficients involve products of the Catalan-Qi numbers, Montes Taurus J. Pure Appl. Math. 1 (2019), no. 2, Article ID MTJPAM-D-19-00007, 1-12.
  • [35] 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, arXiv preprint (2020), available online at https://arxiv.org/abs/2005.13515v1.
  • [36] F. Qi, W.-H. Li, S.-B. Yu, X.-Y. Du, and B.-N. Guo, A ratio of many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 115 (2021), no. 2, Paper No. 39, 14 pages; available online at https://doi.org/10.1007/s13398-020-00988-z.
  • [37] F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of the Catalan-Qi function related to the Catalan numbers, SpringerPlus 5 (2016), Paper No. 1126, 20 pages; available online at https://doi.org/10.1186/s40064-016-2793-1.
  • [38] F. Qi, X.-T. Shi, and F.-F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, Filomat 32 (2018), no. 2, 575?587; available online at https://doi.org/10.2298/FIL1802575Q.
  • [39] F. Qi, X.-T. Shi, F.-F. Liu, and D. V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, J. Appl. Anal. Comput. 7 (2017), no. 3, 857-871; available online at https://doi.org/10.11948/2017054.
  • [40] F. Qi, X.-T. Shi, M. Mahmoud, and F.-F. Liu, The Catalan numbers: a generalization, an exponential representation, and some properties, J. Comput. Anal. Appl. 23 (2017), no. 5, 937-944.
  • [41] F. Qi and Y.-H. Yao, Simplifying coeficients in differential equations for generating function of Catalan numbers, J. Taibah Univ. Sci. 13 (2019), no. 1, 947-950; available online at https://doi.org/10.1080/16583655.2019.1663782.
  • [42] F. Qi, Q. Zou, and B.-N. Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Appl. Anal. Discrete Math. 13 (2019), no. 2, 518-541; available online at https://doi.org/10.2298/AADM190118018Q.
  • [43] S. Roman, An Introduction to Catalan Numbers, with a foreword by Richard Stanley, Compact Textbook in Mathematics, Birkhäuser/Springer, Cham, 2015; available online at https://doi.org/10.1007/978-3-319-22144-1.
  • [44] M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2019; available online at https://doi.org/10.1201/9781351215824.
  • [45] R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book, University of Florence, Italy, 2006.
  • [46] R. Sprugnoli, Sums of reciprocals of the central binomial coeficients, Integers 6 (2006), A27, 18 pp.
  • [47] R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015; available online at https://doi.org/10. 1017/CBO9781139871495.
  • [48] H. S. Wilf, generating functionology, Third edition. A K Peters, Ltd., Wellesley, MA, 2006.
  • [49] R. Witula, E. Hetmaniok, D. Sota, and N. Gawroska, Convolution identities for central binomial numbers, Int. J. Pure App. Math. 85 (2013), no. 1, 171-178; available online at https://doi.org/10.12732/ijpam.v85i1.14.
  • [50] B. Zhang and C.-P. Chen, Sharp Wilker and Huygens type inequalities for trigonometric and inverse trigonometric functions, J. Math. Inequal. 14 (2020), no. 3, 673-684; available online at https://doi.org/10.7153/jmi-2020-14-43.

Yıl 2021, Cilt: 4 Sayı: 1, 57 - 64, 31.03.2021

Feng Qi , Chao-ping Chen , Dongkyu Lım

https://doi.org/10.53006/rna.867047
Cited By: 8

Öz

Kaynakça

  • [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, New York and Washington, 1972.
  • [2] E. P. Adams and R. L. Hippisley, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922.
  • [3] H. Alzer and G. V. Nagy, Some identities involving central binomial coeficients and Catalan numbers, Integers 20 (2020), Paper No. A59, 17 pages.
  • [4] B. C. Berndt, Ramanujan's Notebooks, Part I, With a foreword by S. Chandrasekhar, Springer-Verlag, New York, 1985; available online at https://doi.org/10.1007/978-1-4612-1088-7.
  • [5] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
  • [6] J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
  • [7] K. N. Boyadzhiev, Series with central binomial coeficients, Catalan numbers, and harmonic numbers. J. Integer Seq. 15 (2012), no. 1, Article 12.1.7, 11 pp.
  • [8] D. M. Bradley, A class of series acceleration formulae for Catalan's constant, Ramanujan J. 3 (1999), no. 2, 159-173; available online at https://doi.org/10.1023/A:1006945407723.
  • [9] T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan and Co., Limited, London, 1908.
  • [10] J. M. Campbell, New series involving harmonic numbers and squared central binomial coeficients, Rocky Mountain J. Math. 49 (2019), no. 8, 2513-2544; available online at https://doi.org/10.1216/RMJ-2019-49-8-2513.
  • [11] H. Chen, Interesting series associated with central binomial coeficients, Catalan numbers and harmonic numbers, J. Integer Seq. 19 (2016), no. 1, Article 16.1.5, 11 pp.
  • [12] C.-P. Chen, Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions, Integral Transforms Spec. Funct. 23 (2012), no. 12, 865-873; available online at https://doi.org/10.1080/10652469.2011.644851.
  • [13] P. P. Dályay, Y. J. Ionin, O. P. Lossers, and J. H. Smith, A product of Catalan numbers, Amer. Math. Monthly 125 (2018), no. 1, 86-87; available online at https://doi.org/10.1080/00029890.2018.1397465.
  • [14] A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B 699 (2004), no. 1-2, 3-64; available online at https://doi.org/10.1016/j.nuclphysb.2004.08.020.
  • [15] A. I. Davydychev and M. Yu. Kalmykov, New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams, Nuclear Phys. B 605 (2001), no. 1-3, 266-318; available online at https://doi.org/10.1016/S0550-3213(01) 00095-5.
  • [16] J. Edwards, Differential Calculus, 2nd ed., Macmillan, London, 1982.
  • [17] M. Garcia-Armas and B. A. Seturaman, A note on the Hankel transform of the central binomial coeficients, J. Integer Seq. 11(2008), Article 08.5.8, 9 pages.
  • [18] I. S. Gradshteyn and I. M. Ryzhik, 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, Elsevier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
  • [19] R. P. Grimaldi, Fibonacci and Catalan Numbers, John Wiley & Sons, Inc., Hoboken, NJ, 2012; available online at https: //doi.org/10.1002/9781118159743.
  • [20] C.-Y. Gu and V. J. W. Guo, Proof of two conjectures on supercongruences involving central binomial coeficients, Bull. Aust. Math. Soc. 102 (2020), no. 3, 360-364; available online at https://doi.org/10.1017/s0004972720000118.
  • [21] B.-N. Guo, D. Lim, and F. Qi, Maclaurin series expansions for powers of inverse (hyperbolic) sine, for powers of inverse (hyperbolic) tangent, and for incomplete gamma functions, with applications to second kind Bell polynomials and generalized logsine function, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.10686v5.
  • [22] M. Yu. Kalmykov and A. Sheplyakov, lsjka C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions, Computer Phys. Commun. 172 (2005), no. 1, 45-59; available online at https://doi.org/10.1016/j. cpc.2005.04.013.
  • [23] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009.
  • [24] D. H. Lehmer, Interesting series involving the central binomial coeficient, Amer. Math. Monthly 92 (1985), no. 7, 449-457; available online at http://dx.doi.org/10.2307/2322496.
  • [25] W.-H. Li, F. Qi, O. Kouba, and I. Kaddoura, A further generalization of the Catalan numbers and its explicit formula and integral representation, OSF Preprints (2020), available online at https://doi.org/10.31219/osf.io/zf9xu.
  • [26] M. Mahmoud and F. Qi, Three identities of the Catalan-Qi numbers, Mathematics 4 (2016), no. 2, Article 35, 7 pages; available online at https://doi.org/10.3390/math4020035.
  • [27] J. Miki¢, On certain sums divisible by the central binomial coeficient, J. Integer Seq. 23 (2020), no. 1, Art. 20.1.6, 22 pages.
  • [28] F. Qi, Some properties of the Catalan numbers, Ars Combin. (2022), in press; available online at https://www. researchgate.net/publication/328891537.
  • [29] F. Qi and P. Cerone, Some properties of the Fuss-Catalan numbers, Mathematics 6 (2018), no. 12, Article 277, 12 pages; available online at https://doi.org/10.3390/math6120277.
  • [30] F. Qi, C.-P. Chen, and D. Lim, Five identities involving the product or ratio of two central binomial coeficients, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.02027v1
  • [31] F. Qi, C.-P. Chen, and D. Lim, Several combinatorial identities derived from series expansions of powers of arcsine, arXiv preprint (2021), available online at https://arxiv.org/abs/2101.02027v2.
  • [32] F. Qi and B.-N. Guo, From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions, J. Math. Anal. Appl. 493 (2021), no. 1, Article 124478, 19 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124478.
  • [33] F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (2017), no. 3, Article 40, 31 pages; available online at https://doi.org/10.3390/math5030040.
  • [34] F. Qi and B.-N. Guo, Sums of infinite power series whose coeficients involve products of the Catalan-Qi numbers, Montes Taurus J. Pure Appl. Math. 1 (2019), no. 2, Article ID MTJPAM-D-19-00007, 1-12.
  • [35] 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, arXiv preprint (2020), available online at https://arxiv.org/abs/2005.13515v1.
  • [36] F. Qi, W.-H. Li, S.-B. Yu, X.-Y. Du, and B.-N. Guo, A ratio of many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 115 (2021), no. 2, Paper No. 39, 14 pages; available online at https://doi.org/10.1007/s13398-020-00988-z.
  • [37] F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of the Catalan-Qi function related to the Catalan numbers, SpringerPlus 5 (2016), Paper No. 1126, 20 pages; available online at https://doi.org/10.1186/s40064-016-2793-1.
  • [38] F. Qi, X.-T. Shi, and F.-F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, Filomat 32 (2018), no. 2, 575?587; available online at https://doi.org/10.2298/FIL1802575Q.
  • [39] F. Qi, X.-T. Shi, F.-F. Liu, and D. V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, J. Appl. Anal. Comput. 7 (2017), no. 3, 857-871; available online at https://doi.org/10.11948/2017054.
  • [40] F. Qi, X.-T. Shi, M. Mahmoud, and F.-F. Liu, The Catalan numbers: a generalization, an exponential representation, and some properties, J. Comput. Anal. Appl. 23 (2017), no. 5, 937-944.
  • [41] F. Qi and Y.-H. Yao, Simplifying coeficients in differential equations for generating function of Catalan numbers, J. Taibah Univ. Sci. 13 (2019), no. 1, 947-950; available online at https://doi.org/10.1080/16583655.2019.1663782.
  • [42] F. Qi, Q. Zou, and B.-N. Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Appl. Anal. Discrete Math. 13 (2019), no. 2, 518-541; available online at https://doi.org/10.2298/AADM190118018Q.
  • [43] S. Roman, An Introduction to Catalan Numbers, with a foreword by Richard Stanley, Compact Textbook in Mathematics, Birkhäuser/Springer, Cham, 2015; available online at https://doi.org/10.1007/978-3-319-22144-1.
  • [44] M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2019; available online at https://doi.org/10.1201/9781351215824.
  • [45] R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book, University of Florence, Italy, 2006.
  • [46] R. Sprugnoli, Sums of reciprocals of the central binomial coeficients, Integers 6 (2006), A27, 18 pp.
  • [47] R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015; available online at https://doi.org/10. 1017/CBO9781139871495.
  • [48] H. S. Wilf, generating functionology, Third edition. A K Peters, Ltd., Wellesley, MA, 2006.
  • [49] R. Witula, E. Hetmaniok, D. Sota, and N. Gawroska, Convolution identities for central binomial numbers, Int. J. Pure App. Math. 85 (2013), no. 1, 171-178; available online at https://doi.org/10.12732/ijpam.v85i1.14.
  • [50] B. Zhang and C.-P. Chen, Sharp Wilker and Huygens type inequalities for trigonometric and inverse trigonometric functions, J. Math. Inequal. 14 (2020), no. 3, 673-684; available online at https://doi.org/10.7153/jmi-2020-14-43.
Toplam 50 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Feng Qi 0000-0001-6239-2968

Chao-ping Chen

Dongkyu Lım 0000-0002-0928-8480

Yayımlanma Tarihi 31 Mart 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 4 Sayı: 1

Kaynak Göster

APA Qi, F., Chen, C.-p., & Lım, D. (2021). Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. Results in Nonlinear Analysis, 4(1), 57-64. https://doi.org/10.53006/rna.867047
AMA Qi F, Chen Cp, Lım D. Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. RNA. Mart 2021;4(1):57-64. doi:10.53006/rna.867047
Chicago Qi, Feng, Chao-ping Chen, ve Dongkyu Lım. “Several Identities Containing Central Binomial Coefficients and Derived from Series Expansions of Powers of the Arcsine Function”. Results in Nonlinear Analysis 4, sy. 1 (Mart 2021): 57-64. https://doi.org/10.53006/rna.867047.
EndNote Qi F, Chen C-p, Lım D (01 Mart 2021) Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. Results in Nonlinear Analysis 4 1 57–64.
IEEE F. Qi, C.-p. Chen, ve D. Lım, “Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function”, RNA, c. 4, sy. 1, ss. 57–64, 2021, doi: 10.53006/rna.867047.
ISNAD Qi, Feng vd. “Several Identities Containing Central Binomial Coefficients and Derived from Series Expansions of Powers of the Arcsine Function”. Results in Nonlinear Analysis 4/1 (Mart 2021), 57-64. https://doi.org/10.53006/rna.867047.
JAMA Qi F, Chen C-p, Lım D. Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. RNA. 2021;4:57–64.
MLA Qi, Feng vd. “Several Identities Containing Central Binomial Coefficients and Derived from Series Expansions of Powers of the Arcsine Function”. Results in Nonlinear Analysis, c. 4, sy. 1, 2021, ss. 57-64, doi:10.53006/rna.867047.
Vancouver Qi F, Chen C-p, Lım D. Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function. RNA. 2021;4(1):57-64.

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