A Further Extension for Ramanujan’s Beta Integral and Applications

Gao-Wen Xi and Qiu-Ming Luo
Additional contact information
Gao-Wen Xi: College of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing 401331, China
Qiu-Ming Luo: Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing 401331, China

Mathematics, 2019, vol. 7, issue 2, 1-10

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.

Keywords: q -series; q -exponential operator; q -binomial theorem; q -Gauss formula; q -gamma function; gamma function; Ramanujan’s beta integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/7/2/118/pdf (application/pdf)
https://www.mdpi.com/2227-7390/7/2/118/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:7:y:2019:i:2:p:118-:d:200174

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().