# 多伽玛函数

${\displaystyle {\boldsymbol {m}}}$阶多伽玛函数伽玛函数的第${\displaystyle ({\boldsymbol {m+1}})}$对数导数

${\displaystyle \psi ^{(m)}(\zeta )=\left({\frac {d}{d\zeta }}\right)^{m}\psi (\zeta )=\left({\frac {d}{d\zeta }}\right)^{m+1}\ln \Gamma (\zeta )}$

${\displaystyle \psi (\zeta )=\psi ^{(0)}(\zeta )={\frac {\Gamma '(\zeta )}{\Gamma (\zeta )}}}$

 ${\displaystyle \ln \Gamma (\zeta )\!}$ ${\displaystyle \psi ^{(0)}(\zeta )\!}$ ${\displaystyle \psi ^{(1)}(\zeta )\!}$ ${\displaystyle \psi ^{(2)}(\zeta )\!}$ ${\displaystyle \psi ^{(3)}(\zeta )\!}$ ${\displaystyle \psi ^{(4)}(\zeta )\!}$

## 积分表示法

${\displaystyle \psi ^{(m)}(\zeta )=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-\zeta t}}{1-e^{-t}}}dt}$

## 递推关系

${\displaystyle \psi ^{(m)}(z+1)=\psi ^{(m)}(z)+(-1)^{m}\;m!\;z^{-(m+1)}.}$

## 乘法定理

${\displaystyle k^{m}\psi ^{(m-1)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m-1)}\left(z+{\frac {n}{k}}\right)}$

${\displaystyle k(\psi (kz)-\log(k))=\sum _{n=0}^{k-1}\psi \left(z+{\frac {n}{k}}\right)}$

## 级数表示法

${\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\;m!\;\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}}$

m > 0和任何不等于负数的复数z都成立。还可以用赫尔维茨ζ函数来表示：

${\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\;m!\;\zeta (m+1,z).}$

## 泰勒级数

z = 1时，泰勒级数为：

${\displaystyle \psi ^{(m)}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}(m+k)!\;\zeta (m+k+1)\;{\frac {z^{k}}{k!}},}$