Asymptotics and closed form of a generalized incomplete gamma function

We consider the asymptotic behavior of the function Γ(α,x;b) = ∫x∞ tα−1e−t−btdt, x > 0, α > 0, b ⩾ 0, as x tends to infinity. We give several expansions for the case that α and b are fixed and we give a uniform expansion in which α and b may range through unbounded intervals. We give a closed form for half-integer values of α.