Value of polygamma at positive integer

Theorem

The following formula holds for $n=1,2,\ldots$: $$\psi^{(m)}(n+1)=(-1)^m m! \left[ -\zeta(m+1)+1 + \dfrac{1}{2^{m+1}}+\ldots + \dfrac{1}{n^{m+1}} \right],$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta(m+1)$ denotes the Riemann zeta.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.3$