Certain Classes of the Incomplete $I$-Functions and Their Properties
DOI:
https://doi.org/10.5890/DNC.2023.06.014Abstract
Our current research is motivated by the new interesting generalization (Srivastava et al \cite{1}) of a couple of contour-type Mellin-Barnes integral representations of their incomplete $H$-functions $\gamma^{m,\;n}_{p,\;q}(z)$ and $\Gamma^{m,\;n}_{p,\;q}(z)$, and incomplete $\overline{H}$-functions $\overline{\gamma}^{m,\;n}_{p,\;q}(z)$ and $\overline{\Gamma}^{m,\;n}_{p,\;q}(z)$. By virtue of the gamma functions of incomplete type, that is $\gamma(s,x)$ and $ \Gamma(s,x)$, we introduced here a class of the incomplete $I$-functions $^{\gamma}I^{m,\;n}_{p,\;q}(z)$ and $^{\Gamma}I^{m,\;n}_{p,\;q}(z)$ which leads to a natural extension of a class of $I$-functions. The aim of the present insvestigation is to analyze and examine some impressive properties of these incomplete $I$-functions, inclusive of formulas for decomposition, reduction, derivative and several integral transformations, etc. Further, as the application of newly defined functions, we also formulate and solve a generalized fractional kinetic equation in terms of these incomplete $I$-functions. For the corresponding incomplete $\overline{I}$-functions, we demonstrate the simply determinable extensions of the outcome shown here which also hold. We also raising these effects in certain useful specific forms and established results as well.References
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