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

Abstract

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 $\gamma_*(\alpha,x_-)$ is introduced. Further the polygamma functions $\psi^{(n)}(x)$ are defined for negative integers via the neutrix setting.