Value of polygamma at 1

Theorem

The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta$ denotes the Riemann zeta function.

Reference

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.4.2