New generalized special functions with two generalized M-series at their kernels and solution of fractional PDEs via double Laplace transform

In this paper, we introduce three types of generalized special functions: beta, Gauss hypergeometric, and confluent hypergeometric, all involving two generalized M-series at their kernels. We then give several properties of these functions, such as integral representations, functional relations, summation relations, derivative formulas, transformation formulas, and double Laplace transforms. Furthermore, we obtain solutions of fractional partial differential equations involving these new generalized special functions and then we present graphs of the approximate behavior of the solutions. Also, we introduce a new generalized beta distribution and incomplete beta function. Finally, we establish relationships between the new generalized special functions and other generalized special functions found in the literature.