# §8.8 Recurrence Relations and Derivatives

 8.8.1 $\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\gamma\/}\nolimits\!% \left(a,z\right)-z^{a}e^{-z},$ Symbols: $\mathrm{e}$: 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: 6.5.22 Permalink: http://dlmf.nist.gov/8.8.E1 Encodings: TeX, pMML, png See also: Annotations for 8.8
 8.8.2 $\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\Gamma\/}\nolimits\!% \left(a,z\right)+z^{a}e^{-z}.$ Symbols: $\mathrm{e}$: base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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

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 See also: Annotations for 8.8
 8.8.4 $z\mathop{\gamma^{*}\/}\nolimits\!\left(a+1,z\right)=\mathop{\gamma^{*}\/}% \nolimits\!\left(a,z\right)-\frac{e^{-z}}{\mathop{\Gamma\/}\nolimits\!\left(a+% 1\right)}.$
 8.8.5 $\mathop{P\/}\nolimits\!\left(a+1,z\right)=\mathop{P\/}\nolimits\!\left(a,z% \right)-\frac{z^{a}e^{-z}}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)},$
 8.8.6 $\mathop{Q\/}\nolimits\!\left(a+1,z\right)=\mathop{Q\/}\nolimits\!\left(a,z% \right)+\frac{z^{a}e^{-z}}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)}.$

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

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

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

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

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