Article ID: MTJPAM-D-19-00007

Title: SUMS OF INFINITE POWER SERIES WHOSE COEFFICIENTS INVOLVE PRODUCTS OF THE CATALAN–QI NUMBERS


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-19-00007; Volume 1 / Issue 2 / Year 2019, Pages 1-12

Document Type: Research Paper

Author(s): Feng Qi a , Bai-Ni Guo b

aCollege of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China; School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

bSchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, Henan, China

Received: 5 December 2019, Accepted: 10 December 2019, Available online: 28 December 2019.

Corresponding Author: Bai-Ni Guo (Email address: bai.ni.guo@gmail.com)

Full Text: PDF

Abstract

In the paper, the authors discuss some relations between generalized hypergeometric functions and infinite power series whose coefficients are products of the Catalan-Qi numbers, and derive closed expressions for some special power series whose coefficients are products of the Catalan-Qi numbers.

Keywords: Catalan-Qi function, Catalan number, gamma function, Gauss hypergeometric function, generalized hypergeometric function, infinite power series, rising factorial

References:
  1. T. Amdeberhan, X. Guan, L. Jiu, V. H. Moll, C. Vignat, A series involving Catalan numbers: Proofs and demonstrations, Elem. Math. 71 (3), 109-121, 2016. http://dx.doi.org/10.4171/EM/306
  2. W. Chammam, Several formulas and identities related to Catalan–Qi and q-Catalan–Qi numbers, Indian J. Pure Appl. Math. 50 (4), 1039-1048, 2019. https://doi.org/10.1007/s13226-019-0372-1
  3. N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Sci. Petropolitanae 9, 243-251, 1791.
  4. T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009.
  5. T. Koshy, Z.-G. Gao, Convergence of a Catalan series, College Math. J. 43 (2), 141-146, 2012. http://dx.doi.org/10.4169/college.math.j.43.2.141
  6. D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly 92 (7), 449-457, 1985. http://dx.doi.org/10.2307/2322496
  7. M. Mahmoud, F. Qi, Three identities of the Catalan–Qi numbers, Mathematics 4 (2), 35 (7 pages), 2016 https://doi.org/10.3390/math4020035
  8. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010. http://dlmf.nist.gov/
  9. F. Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Probl. Anal. Issues Anal. 7 (25), 104-115, 2018. https://doi.org/10.15393/j3.art.2018.4370
  10. F. Qi, Parametric integrals, the Catalan numbers, and the beta function, Elem. Math. 72 (3), 103-110, 2017. https://doi.org/10.4171/EM/332.
  11. F. Qi, Some properties of the Catalan numbers, Ars Combin. 160, 2022, in press. https://doi.org/10.13140/RG.2.1.4371.6321
  12. F. Qi, P. Cerone, Some properties of the Fuss–Catalan numbers, Mathematics 6 (12), 277 (12 pages), 2018. https://doi.org/10.3390/math6120277
  13. F. Qi, B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (3), 40 (31 pages), 2017. https://doi.org/10.3390/math5030040
  14. F. Qi, B.-N. Guo, Logarithmically complete monotonicity of a function related to the Catalan–Qi function, Acta Univ. Sapientiae Math. 8 (1), 93-102, 2016. http://dx.doi.org/10.1515/ausm-2016-0006
  15. F. Qi, B.-N. Guo, Logarithmically complete monotonicity of Catalan–Qi function related to Catalan numbers, Cogent Math. 3, 1179379 (6 pages), 2016. http://dx.doi.org/10.1080/23311835.2016.1179379
  16. F. Qi, B.-N. Guo, Mathematical Analysis and Applications: Selected Topics, Some properties and generalizations of the Catalan, Fuss, and Fuss–Catalan numbers (Chapter 5) First Edition, Published by John Wiley & Sons, Inc. , Edited by Michael Ruzhansky, Hemen Dutta, and Ravi P. Agarwal, 101-133, 2018. https://doi.org/10.1002/9781119414421.ch5
  17. F. Qi, M. Mahmoud, X.-T. Shi, F.-F. Liu, Some properties of the Catalan–Qi function related to the Catalan numbers, SpringerPlus 5, 1126 (20 pages), 2016. https://doi.org/10.1186/s40064-016-2793-1
  18. F. Qi, D.-W. Niu, D. Lim, B.-N. Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contrib. Discrete Math. 14 (1), in press, 2019. https://doi.org/10.13140/RG.2.2.34679.52640
  19. F. Qi, X.-T. Shi, P. Cerone, A unified generalization of the Catalan, Fuss, and Fuss–Catalan numbers, Math. Comput. Appl. 24 (2), 49 (16 pages), 2019. https://doi.org/10.3390/mca24020049
  20. F. Qi, X.-T. Shi, F.-F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, Filomat 32 (2), 575-587, 2018. https://doi.org/10.2298/FIL1802575Q
  21. F. Qi, X.-T. Shi, M. Mahmoud, F.-F. Liu, Schur-convexity of the Catalan–Qi function related to the Catalan numbers, Tbilisi Math. J. 9 (2), 141-150, 2016. https://doi.org/10.1515/tmj-2016-0026
  22. 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 (5), 937-944, 2017.
  23. F. Qi, Y.-H. Yao, Simplifying coefficients in differential equations for generating function of Catalan numbers, J. Taibah Univ. Sci. 13 (1), 947-950, 2019. https://doi.org/10.1080/16583655.2019.1663782
  24. F. Qi, Q. Zou, B.-N. Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Appl. Anal. Discrete Math. 13 (2), 518-541, 2019. https://doi.org/10.2298/AADM190118018Q
  25. R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015. https://doi.org/10.1017/CBO9781139871495
  26. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. http://dx.doi.org/10.1002/9781118032572
  27. L. Yin, F. Qi, Several series identities involving the Catalan numbers, Trans. A. Razmadze Math. Inst. 172 (3), 466-474, 2018. https://doi.org/10.1016/j.trmi.2018.07.001
  28. Q. Zou, Analogues of several identities and supercongruences for the Catalan–Qi numbers, J. Inequal. Spec. Funct. 7 (4), 235-241, 2016.
  29. Q. Zou, The q-binomial inverse formula and a recurrence relation for the q-Catalan–Qi numbers, J. Math. Anal. 8 (1), 176-182, 2017.