Polygamma multiplication formula

Theorem

The following formula holds for either the pair $\delta=1, m=0$ or $\delta=0, m>0$: $$\psi^{(m)}(nz)=\delta \log(n)+\dfrac{1}{n^{m+1}} \displaystyle\sum_{k=0}^{n-1} \psi^{(n)} \left( z + \dfrac{k}{n} \right),$$ where $\psi^{(m)}$ denotes the polygamma function and $\log$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.4.8$