Summary: Recently F. Al-Musallam
and S. L. Kalla
[Integral Transforms Spec. Funct. 7, 175-190 (1998; Zbl 0922.33001
)] have developed a generalized hypergeometric function occuring in diffraction theory. We define here a new generalized hypergeometric function, using a special case of Wright’s function. Some differential properties, integral representations and special cases are given. Corresponding incomplete generalized gamma function and its complementary function are also defined and some properties are obtained. The hypergeometric function
) is revisited to obtain more properties, including continuous relations, differential formulas and integral representations. Results given earlier by Al-Musallam and Kalla, Kobayashi and some classical results of incomplete gamma functions can be recovered from some results derived in this paper.