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A Brief Survey and an Analytic Generalization of the Catalan Numbers and Their Integral Representations

Author

Listed:
  • Jian Cao

    (School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China)

  • Wen-Hui Li

    (School of Economics, Henan Kaifeng College of Science Technology and Communication, Kaifeng 475001, China)

  • Da-Wei Niu

    (Department of Science, Henan University of Animal Husbandry and Economy, Zhengzhou 450046, China)

  • Feng Qi

    (Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
    Current address: Independent Researcher, Dallas, TX 75252-8024, USA.)

  • Jiao-Lian Zhao

    (School of Mathematics and Statistics, Weinan Normal University, Weinan 714000, China)

Abstract

In the paper, the authors briefly survey several generalizations of the Catalan numbers in combinatorial number theory, analytically generalize the Catalan numbers, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy’s integral formula in the theory of complex functions, and point out potential directions for further study.

Suggested Citation

  • Jian Cao & Wen-Hui Li & Da-Wei Niu & Feng Qi & Jiao-Lian Zhao, 2023. "A Brief Survey and an Analytic Generalization of the Catalan Numbers and Their Integral Representations," Mathematics, MDPI, vol. 11(8), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1870-:d:1123747
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    References listed on IDEAS

    as
    1. Mansour Mahmoud & Feng Qi, 2016. "Three Identities of the Catalan-Qi Numbers," Mathematics, MDPI, vol. 4(2), pages 1-7, May.
    2. Feng Qi & Pietro Cerone, 2018. "Some Properties of the Fuss–Catalan Numbers," Mathematics, MDPI, vol. 6(12), pages 1-11, November.
    3. Feng Qi & Bai-Ni Guo, 2017. "Integral Representations of the Catalan Numbers and Their Applications," Mathematics, MDPI, vol. 5(3), pages 1-31, August.
    Full references (including those not matched with items on IDEAS)

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