§8.8 Recurrence Relations and Derivatives

8.8.1		$$\gamma (a+1,z)=a\gamma (a,z)-z^a{\mathrm{e}}^{-z},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: 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		$$\mathrm{\Gamma }(a+1,z)=a\mathrm{\Gamma }(a,z)+z^a{\mathrm{e}}^{-z}.$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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 $w(a,z)=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, 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		$$z{\gamma }^{*}(a+1,z)={\gamma }^{*}(a,z)-\frac{{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{*}(a,z)$: 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		$$P(a+1,z)=P(a,z)-\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: 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		$$Q(a+1,z)=Q(a,z)+\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: 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 $n=0,1,2,\mathrm{\dots }$,

8.8.7		$$\gamma (a+n,z)={\left(a\right)}_n\gamma (a,z)-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a+n\right)}{\mathrm{\Gamma }\left(a+k+1\right)}{z}^k,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: 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		$$\gamma (a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\gamma (a-n,z)-z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: 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		$$\mathrm{\Gamma }(a+n,z)={\left(a\right)}_n\mathrm{\Gamma }(a,z)+z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a+n\right)}{\mathrm{\Gamma }\left(a+k+1\right)}{z}^k,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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		$$\mathrm{\Gamma }(a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\mathrm{\Gamma }(a-n,z)+z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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		$$P(a+n,z)=P(a,z)-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: 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		$$Q(a+n,z)=Q(a,z)+z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: 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		$$\frac{d}{dz}\gamma (a,z)=-\frac{d}{dz}\mathrm{\Gamma }(a,z)=z^{a-1}{\mathrm{e}}^{-z},$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: 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		$${\frac{\partial }{\partial a}{\gamma }^{*}(a,z)|}_{a=0}=-E_1\left(z\right)-\ln z.$$
ⓘ Symbols: $E_1\left(z\right)$: exponential integral, ${\gamma }^{*}(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 $n=0,1,2,\mathrm{\dots }$,

8.8.15		$$\frac{d^n}{{dz}^n}(z^{-a}\gamma (a,z))={(-1)}^n{z}^{-a-n}\gamma (a+n,z),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\gamma (a,z)$: 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		$$\frac{d^n}{{dz}^n}(z^{-a}\mathrm{\Gamma }(a,z))={(-1)}^n{z}^{-a-n}\mathrm{\Gamma }(a+n,z),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{\Gamma }(a,z)$: 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		$$\frac{d^n}{{dz}^n}({\mathrm{e}}^z\gamma (a,z))={(-1)}^n{\left(1-a\right)}_n{\mathrm{e}}^z\gamma (a-n,z),$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: 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		$$\frac{d^n}{{dz}^n}(z^a{\mathrm{e}}^z{\gamma }^{*}(a,z))=z^{a-n}{\mathrm{e}}^z{\gamma }^{*}(a-n,z),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{*}(a,z)$: 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		$$\frac{d^n}{{dz}^n}({\mathrm{e}}^z\mathrm{\Gamma }(a,z))={(-1)}^n{\left(1-a\right)}_n{\mathrm{e}}^z\mathrm{\Gamma }(a-n,z).$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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