# §8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

## §8.20(i) Large $z$

 8.20.1 $E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p% }\left(z\right)\right),$ $n=1,2,3,\dots$.

As $z\to\infty$

 8.20.2 $E_{p}\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}},$ $|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta$,

and

 8.20.3 $E_{p}\left(z\right)\sim\pm\frac{2\pi i}{\Gamma\left(p\right)}e^{\mp p\pi i}z^{% p-1}+\frac{e^{-z}}{z}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(p\right)_{k}}}{z^% {k}},$ $\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{7}{2}\pi-\delta$,

$\delta$ again denoting an arbitrary small positive constant. Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).

For an exponentially-improved asymptotic expansion of $E_{p}\left(z\right)$ see §2.11(iii).

## §8.20(ii) Large $p$

For $x\geq 0$ and $p>1$ let $x=\lambda p$ and define $A_{0}(\lambda)=1$,

 8.20.4 $A_{k+1}(\lambda)=(1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\frac{\mathrm{d% }A_{k}(\lambda)}{\mathrm{d}\lambda},$ $k=0,1,2,\dots$,

so that $A_{k}(\lambda)$ is a polynomial in $\lambda$ of degree $k-1$ when $k\geq 1$. In particular,

 8.20.5 $\displaystyle A_{1}(\lambda)$ $\displaystyle=1,$ $\displaystyle A_{2}(\lambda)$ $\displaystyle=1-2\lambda,$ $\displaystyle A_{3}(\lambda)$ $\displaystyle=1-8\lambda+6\lambda^{2}.$ ⓘ Symbols: $A_{k}(\lambda)$ Permalink: http://dlmf.nist.gov/8.20.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 8.20(ii), 8.20 and 8

Then as $p\to\infty$

 8.20.6 $E_{p}\left(\lambda p\right)\sim\frac{e^{-\lambda p}}{(\lambda+1)p}\sum_{k=0}^{% \infty}\frac{A_{k}(\lambda)}{(\lambda+1)^{2k}}\frac{1}{p^{k}},$

uniformly for $\lambda\in[0,\infty)$.

For further information, including extensions to complex values of $x$ and $p$, see Temme (1994b, §4) and Dunster (1996b, 1997).