# §8.8 Recurrence Relations and Derivatives

 8.8.1 $\gamma\left(a+1,z\right)=a\gamma\left(a,z\right)-z^{a}e^{-z},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $a$: parameter A&S Ref: 6.5.22 Permalink: http://dlmf.nist.gov/8.8.E1 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8
 8.8.2 $\Gamma\left(a+1,z\right)=a\Gamma\left(a,z\right)+z^{a}e^{-z}.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.19(v) Permalink: http://dlmf.nist.gov/8.8.E2 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

If $w(a,z)=\gamma\left(a,z\right)$ or $\Gamma\left(a,z\right)$, then

 8.8.3 $w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$ ⓘ Symbols: $z$: complex variable, $a$: parameter and $w(a,z)$: function Permalink: http://dlmf.nist.gov/8.8.E3 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8
 8.8.4 $z\gamma^{*}\left(a+1,z\right)=\gamma^{*}\left(a,z\right)-\frac{e^{-z}}{\Gamma% \left(a+1\right)}.$
 8.8.5 $P\left(a+1,z\right)=P\left(a,z\right)-\frac{z^{a}e^{-z}}{\Gamma\left(a+1\right% )},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter A&S Ref: 6.5.21 Referenced by: §8.25(v) Permalink: http://dlmf.nist.gov/8.8.E5 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8
 8.8.6 $Q\left(a+1,z\right)=Q\left(a,z\right)+\frac{z^{a}e^{-z}}{\Gamma\left(a+1\right% )}.$

For $n=0,1,2,\dots$,

 8.8.7 $\gamma\left(a+n,z\right)={\left(a\right)_{n}}\gamma\left(a,z\right)-z^{a}e^{-z% }\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)}{\Gamma\left(a+k+1\right)}z^{k},$
 8.8.8 $\gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \gamma\left(a-n,z\right)-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
 8.8.9 $\Gamma\left(a+n,z\right)={\left(a\right)_{n}}\Gamma\left(a,z\right)+z^{a}e^{-z% }\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)}{\Gamma\left(a+k+1\right)}z^{k},$
 8.8.10 $\Gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \Gamma\left(a-n,z\right)+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
 8.8.11 $P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)},$
 8.8.12 $Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)}.$
 8.8.13 $\frac{\mathrm{d}}{\mathrm{d}z}\gamma\left(a,z\right)=-\frac{\mathrm{d}}{% \mathrm{d}z}\Gamma\left(a,z\right)=z^{a-1}e^{-z},$
 8.8.14 $\left.\frac{\partial}{\partial a}\gamma^{*}\left(a,z\right)\right|_{a=0}=-E_{1% }\left(z\right)-\ln z.$

For $E_{1}\left(z\right)$ see §8.19(i).

For $n=0,1,2,\dots$,

 8.8.15 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\gamma\left(a+n,z\right),$
 8.8.16 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\Gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\Gamma\left(a+n,z\right),$
 8.8.17 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}\gamma\left(a,z\right))=(-1)^{% n}{\left(1-a\right)_{n}}e^{z}\gamma\left(a-n,z\right),$
 8.8.18 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{a}e^{z}\gamma^{*}\left(a,z\right% ))=z^{a-n}e^{z}\gamma^{*}\left(a-n,z\right),$
 8.8.19 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}\Gamma\left(a,z\right))=(-1)^{% n}{\left(1-a\right)_{n}}e^{z}\Gamma\left(a-n,z\right).$