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 ${\text{R}}_C(m,n)$ of Srinivasa Ramanujan [$\text{Mess. Math}$., 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 ${\text{R}}_C(m,n)$ in the form of integrals ${\text{I}}_C^{\ast }(\upsilon ,b,c,\lambda ,y),{\text{J}}_C(\upsilon ,b,c,\lambda ,y),{\text{K}}_C(\upsilon ,b,c,\lambda ,y),{\text{I}}_C(\upsilon ,b,\lambda ,y)$ and solved it in terms of ordinary hypergeometric functions ${}_2{F}_3$, with suitable convergence conditions. Moreover as applications of Ramanujan's integral ${\text{R}}_C(m,n)$, the new nine infinite summation formulas associated with hypergeometric functions ${}_0{F}_1$, ${}_1{F}_2$ and ${}_2{F}_3$ are obtained.

14 pages, 0 figure, Presented in ICSFA-2016