So I've seen the reflection formula for polygamma functions: $\Psi(z) - \Psi(1-z)=-\pi \cot{\pi z}$.
Is there an extension for arguments that sum not to $1$ but to some other (not necessarily integer) constant? In particular, I was hoping to derive a formula for:
$\Psi(\omega p) - \Psi(\omega (1-p))$
where $p \in [0,1]$ and $\omega > 0$.