Some expansions of the exponential integral in series of the incomplete Gamma function

In a recent paper in this journal, Gautschi et al. [W. Gautschi, F.E. Harris, N.M. Temme, Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095-1099] presented an interesting expansion formula for the exponential integral E1(z) in a series of the incomplete Gamma function γ(α,z). Their investigation was motivated by a search for better methods of evaluating the exponential integral E1(z) which occurs widely in applications, most notably in quantum-mechanical electronic structure calculations. The object of the present sequel to the work by Gautschi et al. [Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095-1099] is to give a rather elementary demonstration of the aforementioned expansion formula and to show how easily it can be put in a much more general setting. Some analogous expansion formulas in series of the complementary incomplete Gamma function Γ(α,z) are also considered.