On the representation of gamma-exponential and generalized negative binomial distributions

For more than a century and a half, gamma-type distributions have shown their adequacy in modeling real processes and phenomena. Over time, designs using distributions from the gamma family are becoming more complex in order to improve the applicability of mathematical models to relevant aspects of life. The paper presents a number of results both generalizing and simplifying some classical forms used in the analysis of large-scale and structural mixtures of generalized gamma laws. The gamma-exponential distribution is introduced and its characteristics are described. An explicit form for integral representations of partial probabilities of the generalized negative binomial distribution is given. The results are formulated in terms of the gamma exponential function. The obtained results can be widely used in models that use scale and structural mixtures of distributions with positive unrestricted support to describe processes and phenomena.