  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.
