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

1,2,3,*, 2,4 and 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
PDF [289 KB, uploaded 8 May 2019]

Abstract

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.
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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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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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