Modified Special Functions: Properties, Integral Transforms and Applications to Fractional Differential Equations

Abstract

In this paper, we defined the modified gamma and beta functions, which involving the generalized M-series at their kernels. Then by using the modified beta function we also defined the modified Gauss and confluent hypergeometric functions. Furthermore, we presented some of their properties such that, integral representations, summation formulas and derivative formulas. Also, we applied beta, Mellin, Laplace, Sumudu, Elzaki and general integral transforms to these modified special functions. Moreover, we obtained solutions of fractional differential equations involving the modified special functions, as applications. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.