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Keywords = Fuss–Catalan number

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23 pages, 340 KiB  
Article
Explicit Construction of the Inverse of an Analytic Real Function: Some Applications
by Joaquín Moreno, Miguel A. López and Raquel Martínez
Mathematics 2020, 8(12), 2154; https://doi.org/10.3390/math8122154 - 3 Dec 2020
Cited by 1 | Viewed by 2077
Abstract
In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y=f(x), we provide the power series that defines its inverse [...] Read more.
In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
16 pages, 322 KiB  
Article
A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers
by Feng Qi, Xiao-Ting Shi and Pietro Cerone
Math. Comput. Appl. 2019, 24(2), 49; https://doi.org/10.3390/mca24020049 - 8 May 2019
Cited by 9 | Viewed by 4421
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 [...] Read more.
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. Full article
11 pages, 274 KiB  
Article
Some Properties of the Fuss–Catalan Numbers
by Feng Qi and Pietro Cerone
Mathematics 2018, 6(12), 277; https://doi.org/10.3390/math6120277 - 24 Nov 2018
Cited by 14 | Viewed by 4877
Abstract
In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the [...] Read more.
In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers. Full article
(This article belongs to the Special Issue Special Functions and Applications)
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