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Mathematics 2019, 7(2), 118; https://doi.org/10.3390/math7020118

A Further Extension for Ramanujan’s Beta Integral and Applications

1
College of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing 401331, China
2
Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing 401331, China
*
Author to whom correspondence should be addressed.
Received: 26 December 2018 / Revised: 19 January 2019 / Accepted: 21 January 2019 / Published: 23 January 2019
(This article belongs to the Special Issue Special Functions and Applications)
Full-Text   |   PDF [252 KB, uploaded 2 February 2019]

Abstract

In 1915, Ramanujan stated the following formula 0 t x 1 ( a t ; q ) ( t ; q ) d t = π sin π x ( q 1 x , a ; q ) ( q , a q x ; q ) , where 0 < q < 1 , x > 0 , and 0 < a < q x . The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As some applications, we obtain some new integral formulas of Ramanujan and also show some new representation with gamma functions and q-gamma functions. View Full-Text
Keywords: q-series; q-exponential operator; q-binomial theorem; q-Gauss formula; q-gamma function; gamma function; Ramanujan’s beta integral q-series; q-exponential operator; q-binomial theorem; q-Gauss formula; q-gamma function; gamma function; Ramanujan’s beta integral
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 (CC BY 4.0).
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Xi, G.-W.; Luo, Q.-M. A Further Extension for Ramanujan’s Beta Integral and Applications. Mathematics 2019, 7, 118.

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