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Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 437--454 | DOI:10.5890/DNC.2023.06.014

Kamlesh Jangid$^{1}$, Sanjay Bhatter$^{2}$, Sapna Meena$^{2}$ , Sunil Dutt Purohit$^{1}$

$^{1}$ Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India

$^{2}$ Department of Mathematics, Malaviya National Institute of Technology, Jaipur, India

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Abstract

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.

Acknowledgments

The authors thank the referees for their concrete suggestions which resulted in a better organization of this article.

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