<s>[INST] What is the fractional derivative of the function $\pi \cot (\pi x)$? 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.
 [/INST] Just a comment, but too long...
Maybe this ... Write down the derivative, second derivative, third derivative,... of cot(x) to get a sequence of polynomials in cot(x).
$$\begin{align}
 \cot(x) &= \cot(x)
\\(d/dx) \cot(x) &= -1-\cot(x)^2
\\(d/dx)^2 \cot(x) &= 2 \cot(x)+2 \cot(x)^3
\\(d/dx)^3 \cot(x) &= -2-8 \cot(x)^2-6 \cot(x)^4
\\(d/dx)^4 \cot(x) &= 16 \cot(x)+40 \cot(x)^3+24 \cot(x)^5
\\(d/dx)^5 \cot(x) &= -16-136 \cot(x)^2-240 \cot(x)^4-120 \cot(x)^6
\\(d/dx)^6 \cot(x) &= 272 \cot(x)+1232 \cot(x)^3+1680 \cot(x)^5+720 \cot(x)^7
\\(d/dx)^7 \cot(x) &= -272-3968 \cot(x)^2-12096 \cot(x)^4-13440 \cot(x)^6-5040 \cot(x)^8
\end{align}$$
RECOGNIZE those polynomials as some previously known orthogonal polynomials. See if there are fractional-index versions for those.
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