Use of the generalized Lommel–Wright function in a unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions

Abstract

Motivated essentially by their importance and usefulness in diffraction theory and probability distributions, several interesting generalizations of the Gamma-type functions were investigated extensively in recent years. In this sequel, to these earlier investigations on the subject, the authors present a systematic study of a unification and generalization of some of these Gamma-type functions, which is defined here by means of certain close relatives of the Fox–Wright hypergeometric function _p Ψ _q (z), including e.g., the generalized Lommel–Wright function . A probability density function associated with the generalized Gamma-type function investigated in this article, together with several other related results in the theory of probability and statistics, are also considered.

Keywords:

• Generalized Lommel–Wright function
• Gamma and incomplete Gamma functions
• Fox–Wright hypergeometric function
• Tricomi function
• Srivastava–Daoust multivariable hypergeometric function
• Diffraction theory
• Wiener–Hopf technique
• Probability density function
• Moment generating function
• Survival (or reliability) function
• Hazard (or failure) rate function
• Mean residual life (or remaining life expectancy) function

Acknowledgements

The present investigation was supported, in part, by CONDES (Consejo de Desarrolo Cientifico y Humanístico) de la Universidad del Zulia and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.