A novel extension of weighted hyper geometric functions and fractional derivative

Abstract

In this paper we present a number of weighted hyper geometric functions and the suitable generalization of Caputo fractional derivation. We look into the viable outcomes to find out arrangements of incomplete differential conditions or differential conditions regarding our outcomes. The parameters of the different Erd'elyi-Kober $(E-K)$ cut off basic fulfill the conditions $\beta_k\left(\gamma_k+1\right)>\frac{\mu}{p}-1, \delta_k>0, k=1, \ldots \ldots, m$, at that point $I_{\left(\beta_k\right), m}^{\left.\left(\gamma_k\right), \delta_k\right)}, f(z)$ exists wherever on top of $(0, \infty)$ plus it is a partial direct administrator. In addition, a number of straight and bilinear relatives are acquired by methods like Mellin Function, $H$-Function, $G$-Function, and Appell Functions for the referenced inference administrator. At that point a portion of the considered hyper geometric functions are resolved as far as the summed up Mittag-Leffler work $E_{\left(\rho_j\right)^\lambda}^{\left(\gamma_j\right)\left(I_j\right)}\left[z_1, z_2, \ldots, z_k\right]$ and the summed up polynomials $S_n^m[x]$. The limit conduct of some different class of weighted hyper geometric functions is portrayed as far as Frost man's $\alpha$-limit. It manages results of driving E-K partial integrals both of structures ( $\mathrm{R}-\mathrm{L}$ type) and their right-hand sided analogs. At long last, a function is known utilizing our partial administrator in the issue of fractional analytics of varieties some out of the ordinary cases of the main result here are also well thought-out.