Polygamma reflection formula

Theorem

The following formula holds: $$(-1)^m \psi^{(m)}(1-z)-\psi^{(m)}(z)=\pi \dfrac{\mathrm{d}^m}{\mathrm{d}z^m} \cot(\pi z),$$ where $\psi^{(m)}$ denotes the polygamma, $\pi$ denotes pi, and $\cot$ denotes the cotangent.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.4.7