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.
In this paper, we introduce a new generalization of Aradhana distribution called as Weighted Aradhana Distribution (WID). The statistical properties of this distribution are derived and the model parameters are estimated by maximum likelihood estimation. Simulation study of ML estimates of the parameters is carried out in R software. Finally, an application to real data set is presented to examine the significance of newly introduced model.
In this paper, we obtain a (p, v)-extension of the Appell hypergeometric function F 1 (·), together with its integral representation, by using the extended Beta function. Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell functionF 1,p,v (·) involving Meijer's G-function are obtained.MSC: 33C60, 33C65, 33B15, 33C45, 33C10
In this paper, we obtain a (p, ν)-extension of Srivastava's triple hypergeometric function H B (·), together by using the extended Beta function B p,ν (x, y) introduced in [19]. We give some of the main properties of this extended function, which include several integral representations involving Exton's hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.MSC: 33C60, 33C65, 33C70, 33B15, 33C05, 33C45, 33C10
We obtain a (p,?)-extension of Srivastava?s triple hypergeometric function HC(?) by employing the extended Beta function Bp,?(x, y) introduced in Parmar et al. [J. Class. Anal. 11 (2017), 91-106]. We give some of the main properties of this extended function, which include several integral representations, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.
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