Generalisation of $\Gamma$ function?

Question

Let the incomplete $\Gamma$-function be defined as: \begin{equation} \Gamma(c+1,x)=\int^\infty_x t^ce^{-t}dt \end{equation} Is there a well studied generalisation of the incomplete gamma function that would work when the R.H.S is \begin{equation} \int^\infty_x f(t)\ e^{-t}dt \end{equation}