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

TL;DR

This paper extends the incomplete gamma function and polygamma functions to negative arguments, introducing new definitions and exploring their properties within a neutrix framework.

Contribution

It introduces a novel incomplete gamma type function for negative parameters and defines polygamma functions at negative integers using neutrix calculus.

Findings

Extended definitions of incomplete gamma function to negative arguments

Defined polygamma functions for negative integers via neutrix calculus

Provided foundational properties for these extended functions

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.