Fractional calculus for Gauss equation and hipergeometric function

This thesis consists of five chapters. In the first chapter; it is given the history of fractional calculus and Gauss equation, the works of famous mathematicians about this subject. In the second chapter; some definitions, functions and teorems used for fractional calculus for Hypergeometric Gauss equation are mentioned. In the third chapter; there has been mentioned of fractional solution of homogeneous Gauss equation. In the fourth chapter; it has been mentioned for fractional solution of non-homogeneous Gauss equation. In the fifth chapter; fractional calculus of homogeneous Gauss equation is obtained in the form of Hypergeometric function. Key words: Fractional Derivative and Integrals (Differintegrals), Gamma Function, Beta Function, Gauss Equation, Hypergeometric Functions, Leibniz's Rule.