§8.8 Recurrence Relations and Derivatives

Notes:: These results follow straightforwardly from the definitions in §8.2(i).
Keywords:: derivatives, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.8

8.8.1 $\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\gamma\/}\nolimits\!\left(a,z\right)-z^{a}e^{{-z}},$

Symbols:: $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.2 $\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\Gamma\/}\nolimits\!\left(a,z\right)+z^{a}e^{{-z}}.$

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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

If $w(a,z)=\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ or $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ , 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

8.8.4 $z\mathop{\gamma^{{*}}\/}\nolimits\!\left(a+1,z\right)=\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)-\frac{e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.8.E4
Encodings:: TeX, pMML, png

8.8.5 $\mathop{P\/}\nolimits\!\left(a+1,z\right)=\mathop{P\/}\nolimits\!\left(a,z\right)-\frac{z^{a}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : 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

8.8.6 $\mathop{Q\/}\nolimits\!\left(a+1,z\right)=\mathop{Q\/}\nolimits\!\left(a,z\right)+\frac{z^{a}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : 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

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

8.8.7 $\mathop{\gamma\/}\nolimits\!\left(a+n,z\right)=\left(a\right)_{{n}}\mathop{\gamma\/}\nolimits\!\left(a,z\right)-z^{a}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+n\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}z^{k},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.8 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-n\right)}\mathop{\gamma\/}\nolimits\!\left(a-n,z\right)-z^{{a-1}}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-k\right)}z^{{-k}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.9 $\mathop{\Gamma\/}\nolimits\!\left(a+n,z\right)=\left(a\right)_{{n}}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)+z^{a}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+n\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}z^{k},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.10 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-n\right)}\mathop{\Gamma\/}\nolimits\!\left(a-n,z\right)+z^{{a-1}}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-k\right)}z^{{-k}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.11 $\mathop{P\/}\nolimits\!\left(a+n,z\right)=\mathop{P\/}\nolimits\!\left(a,z\right)-z^{a}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : 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

8.8.12 $\mathop{Q\/}\nolimits\!\left(a+n,z\right)=\mathop{Q\/}\nolimits\!\left(a,z\right)+z^{a}e^{{-z}}\sum _{{k=0}}^{{n-1}}\frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : 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

8.8.13 $\frac{d}{dz}\mathop{\gamma\/}\nolimits\!\left(a,z\right)=-\frac{d}{dz}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=z^{{a-1}}e^{{-z}},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.14 $\left.\frac{\partial}{\partial a}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)\right|_{{a=0}}=-\mathop{E_{1}\/}\nolimits\!\left(z\right)-\mathop{\ln\/}\nolimits z.$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.24
Permalink:: http://dlmf.nist.gov/8.8.E14
Encodings:: TeX, pMML, png

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

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

8.8.15 $\frac{{d}^{n}}{{dz}^{n}}(z^{{-a}}\mathop{\gamma\/}\nolimits\!\left(a,z\right))=(-1)^{n}z^{{-a-n}}\mathop{\gamma\/}\nolimits\!\left(a+n,z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.8.E15
Encodings:: TeX, pMML, png

8.8.16 $\frac{{d}^{n}}{{dz}^{n}}(z^{{-a}}\mathop{\Gamma\/}\nolimits\!\left(a,z\right))=(-1)^{n}z^{{-a-n}}\mathop{\Gamma\/}\nolimits\!\left(a+n,z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.8.17 $\frac{{d}^{n}}{{dz}^{n}}(e^{z}\mathop{\gamma\/}\nolimits\!\left(a,z\right))=(-1)^{n}\left(1-a\right)_{{n}}e^{z}\mathop{\gamma\/}\nolimits\!\left(a-n,z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.8.E17
Encodings:: TeX, pMML, png

8.8.18 $\frac{{d}^{n}}{{dz}^{n}}(z^{a}e^{z}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right))=z^{{a-n}}e^{z}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a-n,z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : 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

8.8.19 $\frac{{d}^{n}}{{dz}^{n}}(e^{z}\mathop{\Gamma\/}\nolimits\!\left(a,z\right))=(-1)^{n}\left(1-a\right)_{{n}}e^{z}\mathop{\Gamma\/}\nolimits\!\left(a-n,z\right).$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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