Generalized Gamma, Beta and Hypergeometric Functions Defined by Wright Function and Applications to Fractional Differential Equations

When the literature is examined, it is seen that there are many studies on the generalizations of gamma, beta and hypergeometric functions. In this paper, new types of generalized gamma and beta functions are defined and examined using the Wright function. With the help of generalized beta function, new type of generalized Gauss and confluent hypergeometric functions are obtained. Furthermore, some properties of these functions such as integral representations, derivative formulas, Mellin transforms, Laplace transforms and transform formulas are determined. As examples, we obtained the solution of fractional differential equations involving the new generalized beta, Gauss hypergeometric and confluent hypergeometric functions. Finally, we presented their relationship with other generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions, which can be found in the literature.