Some Remarks on Incomplete Gamma Type Function $\gamma_*(\alpha,x_{-})$

Emin Özcağ, İnci Ege

The incomplete gamma type function $\gamma(\alpha,x_-)$ is defined as locally summable function on the real line for $\alpha>0$ by \begin{align*} \gamma_*(lpha,x_-)&=\begin{cases} ıt_{0}^{x}|u|^{lpha-1}e^{-u}du, & xeq0, 0, & x>0 \end{cases} &=ıt_{0}^{-x_-}|u|^{lpha-1}e^{-u}du \end{align*} the integral divergining $\alpha\leq0$ and by using the recurrence relation \[ \gamma_*(lpha+1,x_-)=-lpha\gamma_*(lpha,x_-)-x^lpha_-e^{-x} \] the definition of $\gamma_*(\alpha,x_-)$ can be extended to the negative non-integer values of $\alpha$. Recently the authors [8] defined $\gamma_*(-m,x_-)$ for $m=0,1,2,\ldots$. In this paper we define the derivatives of the incomplete gamma type function $\gamma_*(\alpha,x_-)$ as a distribution for all $\alpha<0$.