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arXiv · 1805.02148

Generalizations of Ramanujan integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications

M. I. Qureshi·Showkat Ahmad Dar
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Abstract

In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral $\textbf{R}_{C}(m,n)$ in the form of integrals $\textbf{I}^{*}_{C}(\upsilon,b,c,\lambda,y), \textbf{J}_C (\upsilon,b,c,\lambda,y), \textbf{K}_{C} (\upsilon,b,c, \lambda,y), \textbf{I}_{C}(\upsilon,b,\lambda,y)$ and solved it in terms of ordinary hypergeometric functions ${}_2 F_3$, with suitable convergence conditions. Moreover as applications of Ramanujan's integral $\textbf{R}_{C}(m,n)$, the new nine infinite summation formulas associated with hypergeometric functions ${}_{0}F_{1}$, ${}_{1}F_{2}$ and ${}_{2}F_{3}$ are obtained.

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BibTeXRIS

M. I. Qureshi, Showkat Ahmad Dar. 2018-05-06. Generalizations of Ramanujan integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications. https://arxiv.org/abs/1805.02148

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