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I need to know the $n$-th derivative of $\dfrac{1}{-x+x e^{i\pi x}}$ or the n-th derivative of $\dfrac{1}{-1+e^{i\pi x}}$.

Maple gives the answer for the n-th derivative of $\dfrac{1}{-x+x e^{i\pi x}}$: $$\sum\limits_{m=0}^n (-m)_m x^{-m-1} \binom{n}{m} \frac{\partial ^{n-m}}{\partial x^{n-m}}\frac{1}{-1+e^{i \pi x}}$$

I need your help. Thanks

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    Why do you need...? – 2017-01-17
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    for a paper if i get a satisfactory answer i would add your name to the paper. – 2017-01-17
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    No, I have no interest! – 2017-01-17
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    I do not really understand your needs. In your other question you (essentially) asked for the $n$-th derivative of $x\cot(\pi x)$, I gave you the full Taylor series and you commented with something like "not useful, too complicated". Well, your function is essentially a $\cot$, hence I guess the answer you are looking for is "too complicated" also in this case. – 2017-01-17
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    The Taylor coefficients depend on Bernoulli numbers / values of the $\zeta$ function. I am sorry if that is too complicated, but I have not the power to change them. Please see $(17)$ here: http://mathworld.wolfram.com/Cotangent.html – 2017-01-17
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    Dear Jack D'Aurizio you are right the formula (17) seems to be a way to compute it thank you. – 2017-01-17

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If one defines $\enspace\displaystyle \frac{d^k}{dx^k}\frac{1}{e^{ax}-1} := \frac{(-1)^k a^k e^{ax}}{(e^{ax}-1)^{k+1}}\sum\limits_{j=0}^{k-1}a_{k-1,j}\,e^{axj}\enspace$ then one gets the recursion

$a_{k,j} =(k-j+1)a_{k-1,j-1}+(j+1)a_{k-1,j}\enspace$ with $\enspace a_{k-1,k}=a_{k,-1}:=0$ .

The first values are $\enspace a_{0,0}=a_{1,0}=a_{1,1}=1$ , $\enspace a_{2,0}=1$ , $\enspace a_{2,1}=4\enspace $ and $\enspace a_{2,2}=1$ .

This can be put with $\enspace k:=n-m\enspace$ into the sum which Maple gives for the $n^{th}$ derivation.

Note:

If you define $\enspace\displaystyle \frac{d^k}{dx^k}\frac{1}{e^{ax}-1} := \frac{(-1)^k a^k}{(e^{ax}-1)^{k+1}}\sum\limits_{j=0}^k b_{k,j}\,e^{axj}\enspace$ then maybe the sum looks nicer and can be used for $k=0$ too. As an exercise you can determine the recursion of $\enspace b_{k,j}$ .