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

TL;DR

This paper analytically solves an unsolved Ramanujan integral using hypergeometric functions, generalizes it through new integral forms, and derives nine new infinite summation formulas involving hypergeometric functions.

Contribution

It provides the first analytical solution to Ramanujan's integral using hypergeometric methods and introduces several generalized integral forms with explicit hypergeometric solutions.

Findings

Analytical solution of Ramanujan's integral $ extbf{R}_{C}(m,n)$.

Generalizations of Ramanujan's integral in new integral forms.

Nine new infinite summation formulas involving hypergeometric functions.

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.