An Extended Gamma Function Involving a Generalized Hypergeometric Function

The objective of this paper is to define and study new generalized extended gamma functions. A generalized extended gamma probability density function involving generalized hypergeometric function is also defined. Closed form representations of the generalized gamma functions and the moment generating function are derived in the form of H-function using inverse Mellon transform techniques. Incomplete gamma-type functions and some special cases are discussed. Recurrence relations of generalized gamma functions are also discussed. New generalized gamma probability density function represents a unified form of several gamma and inverse Gaussian densities. Numerous well-known gamma-type functions and densities such as gamma, generalize gamma, two parameter Weibull, generalize Weibull, the Rayleigh, half-normal, Maxwell, generalize inverse Gaussian, negative binomial, chi-square, the Erlang distributions can also be obtained as special cases . Some statistical functions of a generalized gamma random variable are also derived. 2010 mathematics subject classification: 33C20 • 33E50 • 60E05