Analytical properties and aspects of computation of the gamma-exponential function

Mixtures of probability distributions play an important role in modern analysis and modeling of complex processes. Traditionally, much research attention is paid to the distributions from the gamma class. The main probabilistic characteristics of scale mixtures of generalized gamma distributions cannot be expressed in elementary functions that complicates the process of analytical research and often leads to unreasonably large computational difficulties. The paper analyzes aspects of computation and analytical properties of the gamma-exponential function which has proven itself as a convenient tool for studying scale mixtures of generalized gamma distributions. The presented results expand the usage of functions like the Mittag-Leffler function in the analysis of probability distributions.