Pathway fractional integral operator and matrix-variate functions

Abstract

A general real matrix-variate pathway fractional integral operator is introduced here, which covers all the matrix-variate fractional integrals and almost all the extended densities for the pathway parameter α<1 and α→1. Through this new fractional integral operator, one can go to matrix-variate gamma to matrix-variate Gaussian or normal density with appropriate parametric values. In the present paper, we bring out the idea of matrix-variate pathway to the corresponding fractional integral transform. Consequently, a scalar version of pathway fractional integral operator can also be deduced, which generalizes the classical Reimann–Liouville fractional integration operator. The scalar case of the pathway operators have found applications in reaction–diffusion problems, non-extensive statistical mechanics, non-linear waves, fractional differential equations, non-stable neighbourhoods of physical system, etc. It is hoped that by working out the mathematics of the corresponding matrix-variate analogues, the existing physical theories of scalar variable cases will be given extensions. It gives a connection to the statistical distribution theory. Thus, a path can be created to go from pathway fractional integral operator to the corresponding Laplace transform through the pathway parameter α.