On certain generalized incomplete gamma functions
\textit{M. A. Chaudhry} and \textit{S. M. Zubair} [J. Comput. Appl. Math. 55, No. 1, 99-124 (1994; Zbl 0833.33002)] have introduced a generalized incomplete gamma function \(\Gamma(\nu,x; z)\) which reduces to the incomplete gamma function when its variable \(z\) vanishes. In this paper, the authors show that \(\Gamma(\nu, x;z)\) may be written essentially as a single Kampé de Fériet function which in turn may be expressed as a linear combination of two incomplete Weber integrals. Then by using the properties of the latter integrals additional representations for \(\Gamma(\nu,x; z)\) are deduced. In particular, it is shown that \(\Gamma(\nu, x;z)\) can be completely determined by a number of modified Bessel functions for all \(\nu\neq 0\) provided the values of the two incomplete Weber integrals are known for \(0< \text{Re }\nu\leq 1\). When \(\nu= 0\), connections between the generalized incomplete gamma function and incomplete Lipschitz-Hankel integrals are derived and existence of such connections with other special functions are indicated.
scientific article; zbMATH DE number 1326403