A unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions

In view of their importance and usefulness in diffraction theory and probability distributions, several generalizations of the Gamma-type functions were investigated extensively in recent years. The main object of this paper is to present a systematic study of a unification and generalization of these Gamma-type functions, which is defined here by means of the Fox-Wright hypergeometric function pΨq. A probability density function associated with the generalized Gamma-type function investigated in this paper, together with several other related results in the theory of probability and statistics, are also considered.