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
- Weisstein, Eric W. "Polygamma Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygammaFunction.html