Polygamma Function in terms of Hurwitz Zeta Function

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Theorem

$\map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$

where:

$\psi_n$ is the polygamma function
$\Gamma$ is the gamma function
$\zeta$ is the Hurwitz zeta function
$z \in \C_{>0}$
$n \in \Z_{\ge 1}$.


Proof

\(\ds \map \psi z\) \(=\) \(\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z}\) Definition of Digamma Function
\(\ds \) \(=\) \(\ds -\gamma + \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 k - \dfrac 1 {z + k - 1} }\) Reciprocal times Derivative of Gamma Function
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d^n} {\d z^n} \map \psi z\) \(=\) \(\ds -\dfrac {\d^n} {\d z^n} \gamma + \dfrac {\d^n} {\d z^n} \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 k - \dfrac 1 {z + k - 1} }\) taking $n$th derivative
\(\ds \leadsto \ \ \) \(\ds \map {\psi_n} z\) \(=\) \(\ds -\dfrac {\d^n} {\d z^n} \sum_{k \mathop = 1}^\infty \dfrac 1 {z + k - 1}\) Definition of Polygamma Function, Derivative of Constant
\(\ds \) \(=\) \(\ds -\dfrac {\d^n} {\d z^n} \sum_{k \mathop = 0}^\infty \dfrac 1 {z + k}\) reindexing $k$ from $1$ to $0$
\(\ds \) \(=\) \(\ds -\sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^n n!} {\paren {z + k}^{n + 1} }\) $n$th Derivative of Reciprocal
\(\ds \) \(=\) \(\ds \paren {-1}^{n + 1} \map \Gamma {n + 1} \sum_{k \mathop = 0}^\infty \dfrac 1 {\paren {z + k}^{n + 1} }\) Gamma Function Extends Factorial and simplification
\(\ds \) \(=\) \(\ds \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}\) Definition of Hurwitz Zeta Function

$\blacksquare$


Sources

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