Abstract
The object of this paper is to derive three unified fractional derivatives formulae for the Saigo-Maeda operators of fractional integration. The first formula deals with the product of a general class of multivariable polynomials and the multivariable Aleph- function. The second concerns the multivariable polynomials and two multivariable Aleph-functions with the help of the Leibniz rule for fractional derivatives. The last relation also implies the product of a class of multivariable polynomials and the multivariable Aleph-function but it is obtained by the application of the first formula twice and it implicates two independents variables instead of one. The polynomials and the functions have their arguments of the type are quite general nature. These formulae, besides being on very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them put in applications. Our findings provide unifications and extensions of some (known and new) results. We shall give several corollaries and particular cases.
Highlights
Introduction and PreliminariesGupta and Jain [5] have studied unified multiple integrals involving the generalized hypergeometric function, class of multivariable polynomials [9] and multivariable H-function [13,14]
The aim of this paper is to establish a general finite multiple integrals about the generalized hypergeometric function, sequence of functions, general class of multivariable polynomials, the series expansion of the A-function [4] and multivariable I-function defined by Prasad [6]
To evaluate the multiple integrals (2.1), we first express the class of multivariable polynomials in series the multivariable A-function in serie, the sequence of functions in series with the help of equations (1.4), (1.6) and (1.1) respectively. we change the order of the multiple series and the Integrals
Summary
Gupta and Jain [5] have studied unified multiple integrals involving the generalized hypergeometric function, class of multivariable polynomials [9] and multivariable H-function [13,14]. The aim of this paper is to establish a general finite multiple integrals about the generalized hypergeometric function, sequence of functions, general class of multivariable polynomials, the series expansion of the A-function [4] and multivariable I-function defined by Prasad [6]. The multivariable I-function of s-variables defined by Prasad [6] generalizes the multivariable H-function defined by Srivastava and Panda [13,14]. This representation of multiple Mellin-Barnes types integral is:. Throughout the present document, we assume that the existence and convergence conditions of the multivariable I-function. The complex numbers are not zero.Throughout this document, we assume the existence and absolute convergence conditions of the multivariable I-function.
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