Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines

Abstract

The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral one can list out almost all the extended densities for the pathway parameter q < 1 and q → 1. Here we bring out the idea of thicker or thinner tailed models associated with a gamma type distribution as a limiting case of pathway operator. Applications of this extended gamma model in Statistical Mechanics, input-output models, and solar spectral irradiance modeling etc are established.

An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

“…(see [2,3,11,18,24,25]). Equation (18) is the superstatistics of Beck and Cohen [3], in the sense of superimposing another distribution or the distribution of x with superimposed distribution of the parameter θ.…”

“…When δ = 0, ρ = 2, Equation (25) reduces to folded standard normal density. We can extend the generalized gamma model associated with Bessel function in Equation (25) by using the pathway model of Mathai [1], when α < 1 we get the extended function as…”

“…Note that in both the cases, when α → 1, we have Equation (25) and hence it can be considered to be an extended form of Equation (25). This model has wide potential applications in the discipline physical science especially in statistical mechanics, see [27,28].…”