Title: Generalizations and Applications of Srinivasa Ramanujan’s Integral RS(m, n) via Hypergeometric Approach and Integral Transforms
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00026; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 216-226
Document Type: Research Paper
Author(s): Mohammad Idris Qureshi a , Showkat Ahmad Dar b
aDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (Central University), New Delhi, 110025, India
bDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (Central University), New Delhi, 110025, India — Post Graduate Department of Mathematics, Govt. Degree College Boys Baramulla, University of Kashmir, Kashmir, 193502, India
Received: 9 September 2020, Accepted: 20 February 2021, Published: 25 April 2021.
Corresponding Author: Showkat Ahmad Dar (Email address: showkatjmi134@gmail.com)
Full Text: PDF
Abstract
In this paper, we obtain analytical solution of an unsolved integral of Srinivasa Ramanujan, using hypergeometric approach, Mellin transforms, Infinite Fourier sine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer’s symbol. Also we have given some generalizations of the Ramanujan’s integral
in the form of integrals
,
,
,
and solved them in terms of ordinary hypergeometric functions
, with suitable convergence conditions. Moreover as applications of Ramanujan’s integral
, the new three infinite summation formulas associated with hypergeometric functions
and
( or cosine, sine and Lommel functions) are obtained.
Keywords: Generalized hypergeometric function, Infinite Fourier sine transforms, Ramanujan’s integrals, Fox-Wright psi hypergeometric function, Mellin transforms, Series decomposition identity, Bounded sequence
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