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

In this paper, we obtain analytical solution of an unsolved integral ${\mathbf{\text{R}}}_C(m,n)$ of Srinivasa Ramanujan [$\mathit{\text{Mess.}}\mathit{\text{Math}}$., 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 ${\mathbf{\text{R}}}_C(m,n)$ in the form of integrals $$\mathbf{\text{I}}$^{*}{C}(\upsilon,b,c,\lambda,y), $\mathbf{\text{J}}$C (\upsilon,b,c,\lambda,y), $\mathbf{\text{K}}${C} (\upsilon,b,c, \lambda,y), $\mathbf{\text{I}}${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 ${\mathbf{\text{R}}}_C(m,n)$, the new nine infinite summation formulas associated with hypergeometric functions ${}_0{F}_1$, ${}_1{F}_2$ and ${}_2{F}_3$ are obtained.