Series for polygamma in terms of Riemann zeta

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Theorem

The following formula holds for $|z|<1$: $$\psi^{(n)}(z+1)=(-1)^{n+1} \displaystyle\sum_{k=n}^{\infty} k! (-1)^{k+1}\zeta(k+1)z^{k-n},$$ where $\psi^{(n)}$ denotes polygamma and $\zeta$ denotes Riemann zeta.

Proof

References