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Keywords = Catalan–Qi number

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16 pages, 322 KB  
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 11 | Viewed by 5362
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 KB  
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 16 | Viewed by 6227
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)
31 pages, 347 KB  
Review
Integral Representations of the Catalan Numbers and Their Applications
by Feng Qi and Bai-Ni Guo
Mathematics 2017, 5(3), 40; https://doi.org/10.3390/math5030040 - 3 Aug 2017
Cited by 46 | Viewed by 11411
Abstract
In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several [...] Read more.
In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers. Full article
7 pages, 223 KB  
Article
Three Identities of the Catalan-Qi Numbers
by Mansour Mahmoud and Feng Qi
Mathematics 2016, 4(2), 35; https://doi.org/10.3390/math4020035 - 26 May 2016
Cited by 28 | Viewed by 7870
Abstract
In the paper, the authors find three new identities of the Catalan-Qi numbers and provide alternative proofs of two identities of the Catalan numbers. The three identities of the Catalan-Qi numbers generalize three identities of the Catalan numbers. Full article
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