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, and then defined the modified Gauss and confluent hypergeometric functions by using the modified beta function. 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 newly defined the modified special functions. Then, we obtained solution of fractional differential equations involving the modified special functions, as examples. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.