Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370.
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370.
Journal reference: Problemy Analiza--Issues of Analysis 2018, 7, 104-115 DOI: 10.15393/j3.art.2018.4370
Cite as:
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370.
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370.
Abstract
In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.
Subject Areas
improper integral; explicit expression; unified expression; beta function; Wallis ratio; integral representation; Catalan number
Copyright:
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.
Received:
22 April 2018
Commenter:
Feng Qi
The commenter has declared there is no conflict of interests.
Comment:
This preprint has been accepted for publication as
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), in press; Available online at https://doi.org/10.15393/j3.art.2018.????
Received:
11 June 2018
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the author
Comment:
This preprint has been accepted for publication as
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, in press; Available online at https://doi.org/10.15393/j3.art.2018.4370
Received:
28 June 2018
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the author
Comment:
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy AnalizaIssues of Analysis 7 (25) (2018), no. 1, 104115; Available online at https://doi.org/10.15393/j3.art.2018.4370
Received:
28 June 2018
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the author
Comment:
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104115; Available online at https://doi.org/10.15393/j3.art.2018.4370
Received:
28 June 2018
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the author
Comment:
This preprint has been formally published as
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370
Received:
28 June 2018
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the author
Comment:
This preprint has been formally published as
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370
Commenter: Feng Qi
The commenter has declared there is no conflict of interests.
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), in press; Available online at https://doi.org/10.15393/j3.art.2018.????
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, in press; Available online at https://doi.org/10.15393/j3.art.2018.4370
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370