Applications of fractional derivatives to hypergeometric functions

This thesis consists of six chapters. The first chapter is devoted to the introduction. In the second chapter, preliminaries, some definitions, lemmas and theorems used in the other chapters are given. In the third chapter, after giving some information concerning the fractional calculus, Riemann-Liouville and Weyl fractional integrals, the fractional Leibniz rule for functions are investigated. The fourth chapter consists of various applications regarding the fractional calculus. In the fifth chapter, g-Jacobi functions defined with the help of the Riemann-Liouville operator are given and some of their properties are studied and compared with the corresponding properties of the classical Jacobi polynomials. Furthermore, fractional expansions of the Gauss hypergeometric functions are given and the differential equation satisfied by these functions known as F-Gauss functions and its various properties are analyzed. In the sixth chapter, Laguerre functions defined by the Riemann-Liouville operator are presented and some of their properties are investigated and compared with the corresponding properties of the Laguerre polynomials. Besides, a fractional generalization of the Kummer function is studied as a solution of a fractional analogue of the Kummer differential equation.