<s>[INST] Show that the unique solution of $P_n(X) = 0$ is $x=i\frac{e^{2i\theta}+1}{e^{2i\theta}-1}=\frac{1}{\tan(\theta)}$ I need to how that when $ k \neq 0$, the unique solution of $P_n(X)=\frac{(X+i)^{2n+1} - (X - i)^{2n+1}}{2i} = 0$ is $ x=i\frac{e^{2i\theta}+1}{e^{2i\theta}-1}=\frac{1}{\tan(\theta)} $ with $\theta=\frac{k\pi}{2n+1} \in ]-\frac{\pi}{2},\frac{\pi}{2}[ \backslash {0}$.
In the previous questions I've shown that there is no solutions when $k=0$, that if $P_n(x) = 0$, then $x$ verify $(\frac{x+i}{x-i})^{2n+1} = 1$, and that $P_n(x) = 0$ when $\frac{x+i}{x-i} = \exp(\frac{2ik\pi}{2n+1})$ with $k$ between $-n$ and $n$.
I don't really know where to start with this question. Do i just need to replace x in the polynomial with what is said to be the unique solution ?
 [/INST] We want the solutions to $$(x+i)^{2n+1} - (x-i)^{2n+1} =0.$$
Assume $x$ is real and $z=x+i = r e^{i\theta}$ so that $z^*=x-i=re^{-i\theta}$ is the conjugate of $z$.
Substituting and rearranging:
$$z^{2n+1} = {z^\ast}^{2n+1}$$
$$e^{i(2n+1)\theta}=e^{-i(2n+1)\theta}$$
$$e^{i2(2n+1)\theta}=1=e^{2\pi k i}$$
so $$\theta = \frac{k\pi}{2n+1}$$
but $z=x+i$ thus $\Im z \ne 0$, so $k\ne 0$ nor is $k$ a multiple of $2n+1.$
$$x=\Re \left(re^{i\theta}\right)=r\cos \theta, \quad { \text{and} }\quad
 1=\Im \left(re^{i\theta}\right)=r\sin \theta,$$
so $r = 1/\sin \theta$ and $$x=\frac{\cos \theta}{\sin \theta}=\cot \theta=\frac{1}{\tan \theta} =i \frac{e^{i\theta} + e^{-i\theta}}{e^{i\theta} - e^{-i\theta}}=i\frac{e^{2i\theta}+1}{e^{2i\theta}-1}.$$
One indexing is $k\in \{1,2,\cdots, 2n\}.$
The solution is not unique.  $P_n$ is a polynomial of order $2n$, and $P_n(x)=0$ has $2n$ solutions.
$$\begin{aligned} P_n(x)&= \frac{1}{2i}\left[(x+i)^{2n+1}-(x-i)^{2n+1}\right]\\&=\frac{1}{2i} \left[ \sum_{k=0}^{2n+1} \pmatrix{2n+1\\k}x^k i^{2n+1-k}-\sum_{k=0}^{2n+1} \pmatrix{2n+1\\k}x^k (-i)^{2n+1-k}\right]\\ &=\sum_{k=0}^{n} \pmatrix{2n+1\\2k}x^{2k} (-1)^{k+n}\\&=(2n+1)x^{2n}-\pmatrix{2n+1\\2n-2}x^{2n-2} + \cdots +(-1)^n.\end{aligned}$$
Please see a continuation of the discussion for further developments.
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