Defining Incomplete Gamma Type Function with Negative Arguments and Polygamma functions ${\psi }^{(n)}(-m)$

In this paper the incomplete gamma function $\gamma (\alpha ,x)$ and its derivative is considered for negative values of $\alpha$ and the incomplete gamma type function ${\gamma }_{\ast }(\alpha ,x_{-})$ is introduced. Further the polygamma functions ${\psi }^{(n)}(x)$ are defined for negative integers via the neutrix setting.