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DLMF: §8.8 Recurrence Relations and Derivatives ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions

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8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.7 Series Expansions 8.9 Continued Fractions

§8.8 Recurrence Relations and Derivatives

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Keywords:: derivatives, incomplete gamma functions, recurrence relations
Notes:: These results follow straightforwardly from the definitions in §8.2(i).
Permalink:: http://dlmf.nist.gov/8.8
See also:: Annotations for Ch.8

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		$z\gamma^{}\left(a+1,z\right)=\gamma^{}\left(a,z\right)-\frac{e^{-z}}{\Gamma% \left(a+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.8.E4 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

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		$Q\left(a+1,z\right)=Q\left(a,z\right)+\frac{z^{a}e^{-z}}{\Gamma\left(a+1\right% )}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable and $a$ : parameter Referenced by: §8.25(v), §8.4 Permalink: http://dlmf.nist.gov/8.8.E6 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

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

8.8.7		$\gamma\left(a+n,z\right)={\left(a\right)_{n}}\gamma\left(a,z\right)-z^{a}e^{-z% }\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)}{\Gamma\left(a+k+1\right)}z^{k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E7 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.8		$\gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \gamma\left(a-n,z\right)-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E8 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.9		$\Gamma\left(a+n,z\right)={\left(a\right)_{n}}\Gamma\left(a,z\right)+z^{a}e^{-z% }\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)}{\Gamma\left(a+k+1\right)}z^{k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E9 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.10		$\Gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \Gamma\left(a-n,z\right)+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E10 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.11		$P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)},$
ⓘ 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, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E11 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.12		$Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E12 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.13		$\frac{\mathrm{d}}{\mathrm{d}z}\gamma\left(a,z\right)=-\frac{\mathrm{d}}{% \mathrm{d}z}\Gamma\left(a,z\right)=z^{a-1}e^{-z},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.25 Permalink: http://dlmf.nist.gov/8.8.E13 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.14		$\left.\frac{\partial}{\partial a}\gamma^{*}\left(a,z\right)\right\|_{a=0}=-E_{1% }\left(z\right)-\ln z.$
ⓘ Symbols: $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\ln\NVar{z}$ : principal branch of logarithm function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.24 Permalink: http://dlmf.nist.gov/8.8.E14 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

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

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

8.8.15		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\gamma\left(a+n,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E15 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.16		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\Gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\Gamma\left(a+n,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer A&S Ref: 6.5.26 Referenced by: §8.19(v) Permalink: http://dlmf.nist.gov/8.8.E16 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.17		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}\gamma\left(a,z\right))=(-1)^{% n}{\left(1-a\right)_{n}}e^{z}\gamma\left(a-n,z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/8.8.E17 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.18		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{a}e^{z}\gamma^{}\left(a,z\right% ))=z^{a-n}e^{z}\gamma^{}\left(a-n,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer A&S Ref: 6.5.27 Permalink: http://dlmf.nist.gov/8.8.E18 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.8.19		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}\Gamma\left(a,z\right))=(-1)^{% n}{\left(1-a\right)_{n}}e^{z}\Gamma\left(a-n,z\right).$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer Referenced by: §8.19(v) Permalink: http://dlmf.nist.gov/8.8.E19 Encodings: TeX, pMML, png See also: Annotations for §8.8 and Ch.8

8.7 Series Expansions 8.9 Continued Fractions

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