A Study of Fractional Calculus Operators Associated with Generalized Functions

Abstract

Theory of special functions has a long and varied history with immense literature due to their newlineapplications in solving various problems arising in physical, biological and engineering newlinesciences. Special functions have an origin in the solution of partial differential equations newlinesatisfying certain prescribed conditions. At present special functions are defined in several ways newlinenotably by power series, generating functions, infinite products, integrals, difference equations, newlinetrigonometric or orthogonal function series. Eminent mathematicians notably Euler, Legendre, newlineGauss, Jacobi, Weierstrass, Kummer, Riemann, Ramanujan worked hard to develop special newlinefunctions like Bessel functions, Whittaker functions, Gauss hypergeometric function and the newlinepolynomials that go by the names of Jacobi, Legendre, Laguerre, Hermite etc. The Gaussian newlinehyper geometric function pFq and its special cases are commonly used in applied mathematics newlineand mathematical physics. Since pFq diverges for p gtq +1, in an attempt to give meaning to it in newlinethis case , MacRobrtand Meijer introduced the E-function and the G-function respectively. newlineThe main object of this thesis is to obtain numerous applications of fractional derivative newlineoperator concerning special functions by introducing new classes and deriving new properties. newlineOur finding will provide interesting new results and indicate extensions of a number of known newlineresults. In this thesis we investigate a wide class of problems. The third chapter is concerned to newlinethe following points:(i)Recurrence relations of the generalized M-series. (ii)Integral newlinerepresentation and fractional calculus of the generalized K-function.(iii)The functions newlineW(c,u, p,q, x) newlineand W(c,-m, p,q, x) newlinewithin conditions of the generalized K-function and its newlineproperties using fractional integral and differential. we derive recurrence relation of the newlinegeneralized M-series. The outcome is represented in the structure of a theorem. we consider newlinesolutions which are derived as ( ) newline, newlinepMq z newlinea b newlineis the generalized M-series. There are solutions to the newline