Series for polygamma in terms of Riemann zeta

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.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.9$