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

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.