Alpha Function

$\begin{figure}\begin{center}\BoxedEPSF{AlphaFunction.epsf}\end{center}\end{figure}$

$\begin{displaymath} \alpha_n(z) \equiv \int_1^\infty {t^n e^{-zt} dt} = n! z^{-(n+1)}e^{-z}\sum_{k=0}^n {z^k\over k!}. \end{displaymath}$

The alpha function satisfies the Recurrence Relation

$\begin{displaymath} z\alpha_n(z) = e^{-z}+n\alpha_{n-1}(z). \end{displaymath}$