In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.
Motivated by the substantial development in the theory of digamma function, we derive some new identities for the digamma function. These new identities enable us to compute the values of the digamma function for fractional orders in an analogous manner. Also, we mention two errata, found in Jensen’s article (An elementary exposition of the theory of the Gamma function, 1916), and present their correct forms.
The motive of this research paper is to obtain explicit forms of certain extensions and generalizations of Kümmer's third summation theorem, which have not previously appeared in the literature, by using the summation theorem given by Rakha and Rathie (2011). The results derived in this paper are interesting and may be beneficial.
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