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Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 1039–1048 | Cite as

Several formulas and identities related to Catalan-Qi and q-Catalan-Qi numbers

  • Wathek ChammamEmail author
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Abstract

In the paper, the author generalizes several formulas and series identities involving the Catalan numbers and establishes several new formulas and series identities involving the Catalan-Qi numbers and q-Catalan-Qi numbers.

Key words

Formula Catalan numbers Catalan-Qi number q-analogue q-Catalan-Qi number binomial transform hypergeometric series 
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Notes

Acknowledgment

The author is grateful to an anonymous referee for careful corrections to and valuable comments on the original version of this paper. The author would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project No. 125-1440.

References

  1. 1.
    U. Abel, Reciprocal Catalan sums: Solution to problem 11765, Amer. Math. Monthly, 123(4) (2016), 405–406.  https://doi.org/10.4169/amer.math.monthly.123.4.399.Google Scholar
  2. 2.
    G. E. Andrews, Catalan numbers q-Catalan numbers and hypergeometric series, J. Comb. Th. (A), 44 (1987), 267–273.  https://doi.org/10.1016/0097-3165(87)90033-1.MathSciNetzbMATHGoogle Scholar
  3. 3.
    G. E. Andrews, q-Catalan identities, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, (2010), 183–190.zbMATHGoogle Scholar
  4. 4.
    W. Chammam, F. Marcellán, and R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation, Linear Algebra Appl., 436 (2012), 2105–2116.  https://doi.org/10.1016/j.laa.2011.10.010.MathSciNetzbMATHGoogle Scholar
  5. 5.
    K.-W. Chen, Identities from the binomial transform, J. Number Theory, 124(1) (2007), 142–150.  https://doi.org/10.1016/j.jnt.2006.07.015.MathSciNetzbMATHGoogle Scholar
  6. 6.
    I. Gessel, Super ballot numbers, J. Symb. Comput., 14 (1992), 179–194.  https://doi.org/10.1016/0747-7171(92)90034-2.MathSciNetzbMATHGoogle Scholar
  7. 7.
    B-N. Guo and F. Qi, On the Wallis formula, International Journal of Analysis and Applications, 8(1) (2015), 30–38.MathSciNetzbMATHGoogle Scholar
  8. 8.
    C. Krattenthaler, Advanced determinant calculus: A complement, Linear Algebra Appl., 411 (2005), 68–166.  https://doi.org/10.1016/j.laa.2005.06.042.MathSciNetzbMATHGoogle Scholar
  9. 9.
    C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin., 42 (The Andrews Festschrift) (1999), (Article B42q).Google Scholar
  10. 10.
    M. Mahmoud and F. Qi, Three identities of the Catalan-Qi numbers, Mathematics, 4(2) (2016), Article 35.  https://doi.org/10.3390/math4020035.zbMATHGoogle Scholar
  11. 11.
    F. Qi, X.-T. Shi, and F.-F. Liu, Expansions of the exponential and the logarithm of power series and applications, Arab. J. Math., 6(2) (2017), 95–108.  https://doi.org/10.1007/s40065-017-0166-4.MathSciNetzbMATHGoogle Scholar
  12. 12.
    F. Qi, X.-T. Shi, and F.-F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, Filomat, 32(2) (2018), 575–587.  https://doi.org/10.2298/FIL1802575Q.MathSciNetGoogle Scholar
  13. 13.
    F. Qi and B.-N. Guo, Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers, Cogent Mathematics, (2016), 3: 1179379.  https://doi.org/10.1080/23311835.2016.1179379.MathSciNetzbMATHGoogle Scholar
  14. 14.
    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) (2017), 937–944.MathSciNetGoogle Scholar
  15. 15.
    F. Qi and B.-N. Guo, Some properties and generalizations of the Catalan, Fuss, and Fuss-Catalan numbers, Chapter 5 in Mathematical Analysis and Applications: Selected Topics, 101–133; Edited by M. Ruzhansky, H. Dutta, and R. P. Agarwal; Wiley, 2018.  https://doi.org/10.1002/9781119414421.ch5.Google Scholar
  16. 16.
    F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics5(3) (2017), Article 40.  https://doi.org/10.3390/math5030040.zbMATHGoogle Scholar
  17. 17.
    F. Qi and B.-N. Guo, Logarithmically complete monotonicity of a function related to the Catalan-Qi function, Acta Universitatis Sapientiaem Mathematica, 8(1) (2016), 93–102.  https://doi.org/10.1515/ausm-2016-0006.MathSciNetzbMATHGoogle Scholar
  18. 18.
    F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of the Catalan-Qi function related to the Catalan numbers, SpringerPlus (2016), 5: 1126.  https://doi.org/10.1186/s40064-016-2793-1.Google Scholar
  19. 19.
    F. Qi, X.-T. Shi, and B.-N. Guo, Integral representations of the large and little Schröder numbers, Indian J. Pure Appl. Math., 49(1) (2018), 23–38.  https://doi.org/10.1007/s13226-018-0258-7.MathSciNetzbMATHGoogle Scholar
  20. 20.
    F. Qi, X.-T. Shi, M. Mahmoud, and F-F. Liu, Schur-convexity of the Catalan-Qi function related to the Catalan numbers, Tbilisi Mathematical Journal, 9(2) (2016), 141–150.  https://doi.org/10.1515/tmj-2016-0026.MathSciNetzbMATHGoogle Scholar
  21. 21.
    F. Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza-Issues of Analysis, 7(25)(1) (2018), 104–115.  https://doi.org/10.15393/j3.art.2018.4370.MathSciNetGoogle Scholar
  22. 22.
    F. Qi, A. Akkurt, and H. Yildirim, Catalan numbers, k-gamma and k-beta functions, and parametric integrals, J. Computational Analysis and Applications, 25(6) (2018), 1036–1042.MathSciNetGoogle Scholar
  23. 23.
    F. Qi, Parametric integrals, the Catalan numbers, and the beta function, Elemente der Mathematik, 72(3) (2017), 103–110.  https://doi.org/10.4171/EM/332.MathSciNetzbMATHGoogle Scholar
  24. 24.
    C. Radoux, Calcul effectif de certains déterminants de Hankel, Bull. Soc. Math. Belg. Sér. B, 31 (1979), 49–55.MathSciNetzbMATHGoogle Scholar
  25. 25.
    C. Radoux, Déterminant de Hankel construit sur des polynomes liés aux nombres de dérangements, European J. Combin., 12 (1991), 327–329.  https://doi.org/10.1016/S0195-6698(13)80115-1.MathSciNetzbMATHGoogle Scholar
  26. 26.
    L. Yin and F. Qi, Several series identities involving the Catalan numbers, Transactions of A. Razmadze Mathematical Institute, 172(3) (2018), 466–474.  https://doi.org/10.1016/j.trmi.2018.07.001.MathSciNetzbMATHGoogle Scholar
  27. 27.
    Q. Zou, Congruences from q-Catalan identities, Transactions on Combinatorics, 5(4) (2016), 57–67.MathSciNetGoogle Scholar
  28. 28.
    Q. Zou, Analogues of several identities and supercongruences for the Catalan-Qi numbers, J. Inequal. Spec. Funct., 7(4) (2016), 235–241.MathSciNetGoogle Scholar
  29. 29.
    Q. Zou, The q-binomial inverse formula and a recurrence relation for the q-Catalan-Qi numbers, J. Math. Anal., 8(1) (2017), 176–182.MathSciNetGoogle Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of Science Al-ZulfiMajmaah UniversityAl-MajmaahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of Sciences of GabèsGabès University, City of ErriadhGabèsTunisia

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