# Tag Archives: polygamma

## Relation between polygamma and Hurwitz zeta function proof

$\forall \,\, n\geq 1$ $$\psi_{n}(z) \, = \, (-1)^{n+1}n!\,\zeta(n+1,z)$$ $$\textit{proof}$$ Use the series representation of the digamma $$\psi_{0}(z) = -\gamma-\frac{1}{z}+ \sum_{n=1}^\infty\frac{z}{n(n+z)}$$ This can be written as the following $$\psi_{0}(z) = -\gamma + \sum_{k=0}^\infty\frac{1}{k+1}-\frac{1}{k+z}$$ By differentiating with respect to … Continue reading