# §8.9 Continued Fractions

 8.9.1 $\displaystyle\mathop{\Gamma\/}\nolimits\!\left(a+1\right)e^{z}\mathop{\gamma^{% *}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\cfrac{1}{1-\cfrac{z}{a+1+\cfrac{z}{a+2-\cfrac{(a+1)z}{a+3+% \cfrac{2z}{a+4-\cfrac{(a+2)z}{a+5+\cfrac{3z}{a+6-\cdots}}}}}}},$ $a\neq-1,-2,\dots$, Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{e}$: base of exponential function, $\mathop{\gamma^{*}\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $a$: parameter A&S Ref: 6.5.31 (in modified form.) Permalink: http://dlmf.nist.gov/8.9.E1 Encodings: TeX, pMML, png See also: Annotations for 8.9 8.9.2 $\displaystyle z^{-a}e^{z}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\cfrac{z^{-1}}{1+\cfrac{(1-a)z^{-1}}{1+\cfrac{z^{-1}}{1+\cfrac{(% 2-a)z^{-1}}{1+\cfrac{2z^{-1}}{1+\cfrac{(3-a)z^{-1}}{1+\cfrac{3z^{-1}}{1+\cdots% }}}}}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$.

For these expansions and further information see Jones and Thron (1985). See also Cuyt et al. (2008, pp. 240–251).