I know that $\int_0^\infty t^{a-1}(1+t)^{c-a-1}e^{-yt}~dt=\Gamma(a)U(a,c,y)$ , where $\text{Re}(a),\text{Re}(y)>0$ .
How about $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+xt)^{-b}e^{-yt}~dt$ and $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+x_1t)^{-b_1}\cdots(1+x_nt)^{-b_n}e^{-yt}~dt$ , where $\text{Re}(a),\text{Re}(y)>0$ ? Do some definitions in http://en.wikipedia.org/wiki/Humbert_series can help us?