Unified Approach to Fractional Calculus Images Involving the Pathway Transform of Extended k -Gamma Function and Applications

Abstract

A path way model is concerned with the rules of swapping among different classes of functions. Such model is significant to fit a parametric class of distributions for new data. Based on a such model, the path way or $P_{\delta }$ transform is of binomial type and contains a family of different transforms. Taking inspiration from these facts, present research is concerned with the computation of new fractional calculus images involving the extended $k$ -gamma function. The non-integer kinetic equations containing the extended $k$ -gamma function is solved by using pathway transform as well as validated with the earlier obtained results. $P_{\delta }$ transform of Dirac delta function is obtained which proved useful to achieve the purpose. As customary, the results for the frequently used Laplace transform can be recovered by taking $\delta ⟶1$ in the definition of $P_{\delta }$ transform. Important new identities involving the Fox-Wright function are obtained and used to simplify the results. It is remarkable that the several new and novel results involving the classical gamma function became possible by using this approach.