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Open AccessArticle

A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

by Feng Qi 1,2,3,*, Xiao-Ting Shi 2,4 and Pietro Cerone 5
1
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
2
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
3
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
4
Secondary School, Yingxiong Street, Fenyang, Lüliang 032200, China
5
Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC 3086, Australia
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(2), 49; https://doi.org/10.3390/mca24020049
Received: 1 April 2019 / Revised: 30 April 2019 / Accepted: 5 May 2019 / Published: 8 May 2019
In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan–Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization. View Full-Text
Keywords: unified generalization; Catalan number; Fuss number; Fuss–Catalan number; Catalan–Qi function; product-ratio expression; logarithmically complete monotonicity of the second order; integral representation unified generalization; Catalan number; Fuss number; Fuss–Catalan number; Catalan–Qi function; product-ratio expression; logarithmically complete monotonicity of the second order; integral representation
MDPI and ACS Style

Qi, F.; Shi, X.-T.; Cerone, P. A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers. Math. Comput. Appl. 2019, 24, 49.

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