Investigations in certain families of generalized functions of one and several variables and the calculus of arbitrary order

Abstract

first we have considered a more general definition for extended Beta function in the form of series integral representation. For this series integral representation two results are established and their simpler forms are also deduced as the special cases of our main results.Three cases for 1st type of general class of integrals while another three cases for 2nd type of general class of integral are evaluated. newlineThe classes of integrals studied in this chapter are integrals involving the product of several exponential functions and Gauss s hypergeometric function. In our earlier work [61], we have studied a certain general classes of integrals involving exponential function and Gauss s hypergeometric function. newlineThe results established in this chapter are very general Euler type integral formulas. Many known results of our main results are shown to be deduced, while some results are also obtained from the main results established here as new integral formulas involving generating functions and introduced by Bin-Saad [5] in the literature. Certain simpler forms of Beta integrals are also shown to be obtained from our main results which shows the importance of our present study. These fractional order differential equations with their solutions are given in this chapter as the important corollaries newline