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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.

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    You might want to tell us which definition you used to derive the expression; else we won't know which are "other" definitions.2012-04-16
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    As @joriki alludes to, you neglected to mention what definition you're using. The Riemann-Liouville and Caputo definitions sometimes give the same results, but not always.2012-04-17
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    I did not use any particular definition, I derived the formula from completely different considerations (the derivation is only valid for cotangent).2012-04-17
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    I presume you've seen [this](http://functions.wolfram.com/ElementaryFunctions/Cot/20/03/)?2012-04-17
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    Oh, thanks! Very interesting!2012-04-17
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    The first formula I do not understand. What the notation it uses?2012-04-17
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    The second formula uses generalized polygamma of fractional order which itself needs definition.2012-04-17
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    The unusual notation in the first formula is explained [here](http://functions.wolfram.com/Notations/3/).2012-04-17
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    Maybe this ... Write down the derivative, second derivative, third derivative,... of $\cot(x)$ to get a sequence of polynomials in $\cot(x)$. RECOGNIZE those polynomials as some previously known orthogonal polynomials. See if there are fractional-index versions for those.2012-04-17
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    By the way, I wonder why they on Wolfram use their complicated formula if a simpler formula in the closed form exists (on which my formula is based) $(\cot q)^{(p)}=-\frac{1}{\pi^{p+1}}\Gamma(p+1) \zeta(p+1,1-\frac {q}{\pi})-\frac{1}{\pi^{p+1}}\psi^{(p)}(\frac{q}{\pi})$2012-04-17
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    I just added the definition of Balanced Polygamma to the above formula.2012-04-17

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