# §8.8 Recurrence Relations and Derivatives

 8.8.1 $\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\gamma\/}\nolimits\!% \left(a,z\right)-z^{a}e^{-z},$
 8.8.2 $\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)=a\mathop{\Gamma\/}\nolimits\!% \left(a,z\right)+z^{a}e^{-z}.$

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

 8.8.3 $w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$
 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 $\frac{d}{dz}\mathop{\gamma\/}\nolimits\!\left(a,z\right)=-\frac{d}{dz}\mathop{% \Gamma\/}\nolimits\!\left(a,z\right)=z^{a-1}e^{-z},$
 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 $\frac{{d}^{n}}{{dz}^{n}}(z^{-a}\mathop{\gamma\/}\nolimits\!\left(a,z\right))=(% -1)^{n}z^{-a-n}\mathop{\gamma\/}\nolimits\!\left(a+n,z\right),$
 8.8.16 $\frac{{d}^{n}}{{dz}^{n}}(z^{-a}\mathop{\Gamma\/}\nolimits\!\left(a,z\right))=(% -1)^{n}z^{-a-n}\mathop{\Gamma\/}\nolimits\!\left(a+n,z\right),$
 8.8.17 $\frac{{d}^{n}}{{dz}^{n}}(e^{z}\mathop{\gamma\/}\nolimits\!\left(a,z\right))=(-% 1)^{n}\left(1-a\right)_{n}e^{z}\mathop{\gamma\/}\nolimits\!\left(a-n,z\right),$
 8.8.18 $\frac{{d}^{n}}{{dz}^{n}}(z^{a}e^{z}\mathop{\gamma^{*}\/}\nolimits\!\left(a,z% \right))=z^{a-n}e^{z}\mathop{\gamma^{*}\/}\nolimits\!\left(a-n,z\right),$
 8.8.19 $\frac{{d}^{n}}{{dz}^{n}}(e^{z}\mathop{\Gamma\/}\nolimits\!\left(a,z\right))=(-% 1)^{n}\left(1-a\right)_{n}e^{z}\mathop{\Gamma\/}\nolimits\!\left(a-n,z\right).$