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The essentials of fractional calculus according to different approaches thatcan be useful for our applications in the theory of probability and stochasticprocesses are established. In addition to this, from this fractional integralone can list out almost all the extended densities for the pathway parameter $q< 1$ and $q \rightarrow 1$. Here we bring out the idea of thicker or thinnertailed models associated with a gamma type distribution as a limiting case ofpathway operator. Applications of this extended gamma model in StatisticalMechanics, input-output models, and solar spectral irradiance modeling etc areestablished.

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