Abstract:
In the paper we present closed and unified expressions for a sequence of improper integrals in terms of the beta function and the Wallis ratio. Hereafter, we derive integral representations for the Catalan numbers originating from combinatorics.
\Bibitem{Qi18}
\by Qi~Feng
\paper An improper integral, the beta function, the Wallis ratio, and the Catalan numbers
\jour Probl. Anal. Issues Anal.
\yr 2018
\vol 7(25)
\issue 1
\pages 104--115
\mathnet{http://mi.mathnet.ru/pa229}
\crossref{https://doi.org/10.15393/j3.art.2018.4370}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000437686400007}
\elib{https://elibrary.ru/item.asp?id=36509656}
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This publication is cited in the following 10 articles:
Wathek Chammam, Mongia Khlifi, “Several formulas and identities related to the Pochhammer k-symbol and application to a case of the Fuss–Catalan–Qi numbers”, Indian J Pure Appl Math, 2024
Hüseyi̇n Irmak, Tolga Han Açikgöz, “Notes on Various Transforms Identified by Some Special Functions with Complex (or Real) Parameters and Some of Related Implications”, Engineering World, 5 (2023), 108
Wen-Hui Li, Omran Kouba, Issam Kaddoura, Feng Qi, “A further generalization of the Catalan numbers and its explicit formula and integral representation”, Filomat, 37:19 (2023), 6505
S{\i}lay Aytaç YÜKÇÜ, “The Numerical Evaluation Methods for Beta Function”, Süleyman Demirel üniversitesi Fen Edebiyat Fakültesi Fen Dergisi, 17:2 (2022), 288
Ravi Prakash Agarwal, Erdal Karapinar, Marko Kostić, Jian Cao, Wei-Shih Du, “A Brief Overview and Survey of the Scientific Work by Feng Qi”, Axioms, 11:8 (2022), 385
Vito Lampret, “How is the period of a simple pendulum growing with increasing amplitude?”, Mathematica Slovaca, 71:2 (2021), 359
Feng Qi, Xiao-Ting Shi, Pietro Cerone, “A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers”, MCA, 24:2 (2019), 49
Feng Qi, Pshtiwan Othman Mohammed, Jen-Chih Yao, Yong-Hong Yao, “Generalized fractional integral inequalities of Hermite–Hadamard type for (α,m)-convex functions”, J Inequal Appl, 2019:1 (2019)
Wathek Chammam, “Several formulas and identities related to Catalan-Qi and q-Catalan-Qi numbers”, Indian J Pure Appl Math, 50:4 (2019), 1039
Feng Qi, Pietro Cerone, “Some Properties of the Fuss–Catalan Numbers”, Mathematics, 6:12 (2018), 277
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