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

被引：1

作者：

Cao, Jian ^{[1
]}

Li, Wen-Hui ^{[2
]}

Niu, Da-Wei ^{[3
]}

Qi, Feng ^{[4
]}

Zhao, Jiao-Lian ^{[5
]}

机构：

[1] Hangzhou Normal Univ, Sch Math, Hangzhou 311121, Peoples R China

[2] Henan Kaifeng Coll Sci Technol & Commun, Sch Econ, Kaifeng 475001, Peoples R China

[3] Henan Univ Anim Husb & Econ, Dept Sci, Zhengzhou 450046, Peoples R China

[4] Henan Polytech Univ, Inst Math, Jiaozuo 454010, Peoples R China

[5] Weinan Normal Univ, Sch Math & Stat, Weinan 714000, Peoples R China

来源：

MATHEMATICS | 2023年 / 11卷 / 08期

基金：

中国国家自然科学基金;

关键词：

Catalan number; generalized Catalan function; generalized Catalan number; Cauchy's integral formula; generalization; generating function; integral representation; DIFFERENTIAL-EQUATIONS; BELL POLYNOMIALS; IDENTITIES; FORMULAS; QI; INVERSE;

D O I：

10.3390/math11081870

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页数：16

共 50 条

[1] Integral Representations of the Catalan Numbers and Their Applications
Qi, Feng
Guo, Bai-Ni
MATHEMATICS, 2017, 5 (03):
[2] A Generalization of the Catalan Numbers
Kahkeshani, Reza
JOURNAL OF INTEGER SEQUENCES, 2013, 16 (06)
[3] A generalization of Catalan numbers
Vera-Lopez, A.
Garcia-Sanchez, M. A.
Basova, O.
Vera-Lopez, F. J.
DISCRETE MATHEMATICS, 2014, 332 : 23 - 39
[4] A further generalization of the Catalan numbers and its explicit formula and integral representation
Li, Wen -Hui
Kouba, Omran
Kaddoura, Issam
Qi, Feng
FILOMAT, 2023, 37 (19) : 6505 - 6524
[5] CATALAN NUMBERS, THEIR GENERALIZATION, AND THEIR USES
HILTON, P
PEDERSEN, J
MATHEMATICAL INTELLIGENCER, 1991, 13 (02): : 64 - 75
[6] Integral presentations of Catalan numbers
Dana-Picard, Thierry
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY, 2010, 41 (01) : 63 - 69
[7] GENERALIZATION OF A FORMULA OF TOUCHARD FOR CATALAN NUMBERS
GOULD, HW
JOURNAL OF COMBINATORIAL THEORY SERIES A, 1977, 23 (03) : 351 - 353
[8] A Unified Generalization of the Catalan, Fuss, and Fuss-Catalan Numbers
Qi, Feng
Shi, Xiao-Ting
Cerone, Pietro
MATHEMATICAL AND COMPUTATIONAL APPLICATIONS, 2019, 24 (02)
[9] Generalized Bailey's generalization of the Catalan numbers
Frey, DD
Sellers, JA
FIBONACCI QUARTERLY, 2001, 39 (02): : 142 - 148
[10] CUNTZ-KRIEGER ALGEBRAS AND A GENERALIZATION OF CATALAN NUMBERS
Matsumoto, Kengo
INTERNATIONAL JOURNAL OF MATHEMATICS, 2013, 24 (05)

← 1 2 3 4 5 →