A unified study of Fourier series involving the Aleph-function and the Kampé de Fériet's function

Abstract

Recently Yashwant Singh et al [7] have studied Fourier series involving the I-function defined by V.P. Saxena [6]. Motivated by this work, we make an application of an integral involving sine function, exponential function, the product of Aleph-function of one variable and Kampé de Fériet's function. We also evaluate a multiple integral involving the Aleph-function to make its application to derive a multiple exponential Fourier series. Several particular cases are also given at the end.

Application of Jacobi Polynomial and Multivariable Aleph- Function in Heat Conduction in Non-Homogeneous Moving Rectangular Parallelepiped

“…Remark 3.1. For detail and applications of Aleph-function, the reader can refer recent work [2,8,16].…”

Integral Involving the Product of Multivariable Alephfunction, General Class of Srivastava Polynomials and Alephfunction of One Variable

“…The Aleph (ℵ)-function was established by Südland et al [30], but its notation and complete definition are offered below in terms of the Mellin-Barnes type integral (see also, [2,3,7,13,23,25]):…”

Fractional calculus pertaining to multivariable Aleph-function

“…The Aleph-function, introduced by Südland et al [30], however the notation and complete definition is presented here in the following manner in terms of the Mellin-Barnes type integral (see also, [1,2,11,15,26]): for all z different to 0 and…”

“…The Aleph-function of several variables is an extension of the multivariable I-function defined by Sharma and Ahmad [27], itself is an a generalization of G-and H-functions of multiple variables. The multiple Mellin-Barnes integral occurring in this paper will be referred to as the multivariables Alephfunction throughout our present study and given by ℵ (z1, • • • , zr) = ℵ 0, n:m 1 ,n 1 ,••• ,mr ,nr p i ,q i ,τ i ;R:p i (1) , q i (1) ,τ i (1) ;R (1) ;••• ;p i (r) ,q i (r) ;τ i (r) ;R (r)…”