A Study on the Properties of New Generalized Special Functions, their İntegral Transformations, and Applications to Fractional Differential Equations

Abstract

In this paper, we have defined generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions with the help of generalized M-series. Furthermore, we have obtained some properties of these functions, such as integral representations, summation formulas, and derivative formulas. Then we applied beta, Mellin, Laplace, Sumudu, Elzaki, and general integral transformations to new generalized special functions and obtained solutions of fractional differential equations containing new generalized special functions. Finally, we presented the relationship between generalized special functions in the literature and new generalized special functions.