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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