Beta Function (Exponential)

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

Another ``Beta Function'' defined in terms of an integral is the ``exponential'' beta function, given by

$\displaystyle \beta_n(z)$	$\textstyle \equiv$	$\displaystyle \int_{-1}^1 t^ne^{-zt}\,dt$	(1)
	$\textstyle =$	$\displaystyle n!z^{-(n+1)}\left[{e^z\sum_{k=0}^n {(-1)^kz^k\over k!} -e^{-z}\sum_{k=0}^n {z^k\over k!}}\right].$
			(2)

The exponential beta function satisfies the Recurrence Relation

$\begin{displaymath} z\beta_n(z)=(-1)^ne^z-e^{-z}+n\beta_{n-1}(z). \end{displaymath}$

(3)

The first few integral values are

$\displaystyle \beta_0(z)$	$\textstyle =$	$\displaystyle {2\sinh z\over z}$	(4)
$\displaystyle \beta_1(z)$	$\textstyle =$	$\displaystyle {2(\sinh z-z\cosh z)\over z^2}$	(5)
$\displaystyle \beta_2(a)$	$\textstyle =$	$\displaystyle {2(2+z^2)\sinh z-4z\cosh z\over z^3}.$	(6)