This paper is a sequel to a recent work of \textit{N. Virchenko}, \textit{S. L. Kalla} and \textit{A. Al-Zamel} [Integral Transforms Spec Funct. 12, 89-100 (2001; Zbl 1026.33006)] on a generalized hypergeometric function represented in the following integral form: \[ _2R_1(a,b;c; \tau;z)= {\Gamma(c) \over\Gamma (b)\Gamma (c-b)}\int^1_0 t^{b-1} (1-t)^{c-b-1} (1-zt^\tau)^{-a} dt \tag{*} \] Some results on (*) are here established. Furthermore, by using these results the authors also derive several recurrence relations and other properties for certain generalized incomplete gamma function studied by \textit{F. Al-Musallam} and \textit{S. L. Kalla} [Appl. Anal. 66, 173-187 (1997; Zbl 0886.33002); see also same authors, Integral Transforms Spec. Funct. 7, 175-190 (1998; Zbl 0922.33001)] associated with (*) and related to the generalized gamma function \(\Gamma(u,v)\) introduced by \textit{K. Kobayashi} [J. Phys. Soc. Japan 60, 1501-1512 (1991)] occurring in difraction theory. Some special cases referred to the particular value \(\tau=1\) are finally discussed.