What is the fractional derivative of the function $\pi \cot (\pi x)$?
I derived the following expression:
$(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma (-p)}-\Gamma (p+1) \zeta (p+1,1-q)$
where $\psi$ is digamma, $\zeta$ is Hurwitz zeta, $\zeta'$ is the derivative by first argument
I want to know whether it coincides with other, traditional definitions.
This gives the formula $(\cot (q))^{(p)}=-\frac{\zeta'(p+1,\frac q\pi)+(\psi(-p)+\gamma ) \zeta (p+1,\frac q\pi)}{\pi^{p+1}\Gamma (-p)}-\frac 1{\pi^{p+1}}\Gamma (p+1) \zeta (p+1,1-\frac q\pi)$ for fractional derivative of cotangent.