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Probl. Anal. Issues Anal., 2018, Volume 7(25), Issue 1, Pages 104–115 (Mi pa229)  

An improper integral, the beta function, the Wallis ratio, and the Catalan numbers

Qi Fengabc

a Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China
b College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia, 028043, China
c Institute of Mathematics, Henan Polytechnic University, Jiaozuo, Henan, 454010, China

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.

Keywords: improper integral; closed expression; unified expression; beta function; Wallis ratio; integral representation; Catalan number.

DOI: https://doi.org/10.15393/j3.art.2018.4370

Full text: PDF file (288 kB)
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Bibliographic databases:

UDC: 519.58
MSC: Primary 11B65; Secondary 11B75, 11B83, 26A39, 26A42, 33B15
Received: 22.01.2018
Revised: 28.04.2018
Accepted:30.04.2018
Language:

Citation: Qi Feng, “An improper integral, the beta function, the Wallis ratio, and the Catalan numbers”, Probl. Anal. Issues Anal., 7(25):1 (2018), 104–115

Citation in format AMSBIB
\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{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000437686400007}
\elib{http://elibrary.ru/item.asp?id=36509656}


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