On certain generalized incomplete gamma functions

摘要

Recently, Chaudhry and Zubair have introduced a generalized incomplete gamma function Γ(ν, x; z) which reduces to the incomplete gamma function Γ(ν, x) when its variable z vaniable z vanishes. We show that Γ(ν, x; z) may be written essentially as a single Kampe de Feriet function which in turn may be expressed as a linear combination of two incomplete Weber integrals. The by using properties of the latter integrals we deduce additional representations for Γ(ν, x; z). In particular, we show that Γ(ν, x; z) is essentially completely determined by a finite number of modified Bessel functions for all ν ≠ 0 provided we know the values of the two incomplete Weber integrals when 0 < Re ν ≤ 1. When ν = 0 we derive connections between the generalized incomplete gamma function and incomplete Lipschitz-Hankel integrals, and indicate that there exist connections with other special functions.