# Properties of Ultra Gamma Function

## TL;DR

This paper investigates the properties of the Four Parameter Gamma Function, deriving identities, recurrence relations, and connections to other special functions, including hypergeometric functions.

## Contribution

It introduces the Four Parameter Gamma Function, establishes its identities, recurrence relations, and links to classical and p-k Gamma functions, expanding understanding of generalized gamma integrals.

## Key findings

- Derived identities and recurrence relations for the Four Parameter Gamma Function.
- Established relations between the Four Parameter Gamma Function, p-k Gamma Function, and Classical Gamma Function.
- Expressed the Four Parameter Gamma Function in terms of Hypergeometric functions under certain conditions.

## Abstract

In this paper we study the integral of type \[_{\delta,a}\Gamma_{\rho,b}(x) =\Gamma(\delta,a;\rho,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^{\delta}}{a}-\frac{t^{-\rho}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Function. With some conditions we can evaluate Four Parameter Gamma Function in term of Hypergeometric function.