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

Jian Cao, Wen-Hui Li, Da-Wei Niu, Feng Qi () and Jiao-Lian Zhao
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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
Jiao-Lian Zhao: School of Mathematics and Statistics, Weinan Normal University, Weinan 714000, China

Mathematics, 2023, vol. 11, issue 8, 1-16

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.

Keywords: Catalan number; generalized Catalan function; generalized Catalan number; Cauchy’s integral formula; generalization; generating function; integral representation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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