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Let $\zeta$ be an $n$-th root of unity and let $\chi:=\zeta+\zeta^{-1}$. Then $\zeta^k+\zeta^{-k}=P_k(\chi)$ where $P_k\in\Bbb{Z}[X]$ is a polynomial not depending on $n$. For example we have \begin{eqnarray*} \zeta^2+\zeta^{-2}&=&\chi^2-2, \qquad&\text{ so }&\qquad P_2&=&X^2-2,\\ \zeta^3+\zeta^{-3}&=&\chi^3-3\chi, \qquad&\text{ so }&\qquad P_3&=&X^3-3X,\\ \zeta^4+\zeta^{-4}&=&\chi^4-4\chi^2+2, \qquad&\text{ so }&\qquad P_4&=&X^4-4X^2+2,\\ \zeta^5+\zeta^{-5}&=&\chi^5-5\chi^3+5\chi, \qquad&\text{ so }&\qquad P_5&=&X^5-5X^3+5,\\ \zeta^6+\zeta^{-6}&=&\chi^6-6\chi^4+9\chi^2+18, \qquad&\text{ so }&\qquad P_6&=&X^6-6X^4+9X^2+18. \end{eqnarray*} It isn't hard to see that $P_{ab}=P_a\circ P_b=P_b\circ P_a$ for all positive integers $a$ and $b$, and that we have a recurrence relation $$P_a=X^a-\sum_{i=1}\binom{a}{i}P_{a-2i},$$ where we take the convention that $P_k=0$ for all $k<0$, and $P_0=1$. My question is:

Is there a simple explicit expression for $P_k$?

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    $\begingroup$ This may help you: en.wikipedia.org/wiki/Chebyshev_polynomials $\endgroup$ – Seewoo Lee Nov 15 '18 at 16:40
  • $\begingroup$ @SeewooLee Can you turn that comment into an answer? $\endgroup$ – Pedro Tamaroff Nov 15 '18 at 16:43
  • $\begingroup$ @SeewooLee Thank you for the link. So if I understand correctly we have $P_k(X)=2T_k(\frac{X}{2})$, where $T_k$ is the $k$-th Chebyshev polynomial of the first kind? $\endgroup$ – Servaes Nov 15 '18 at 16:46
  • $\begingroup$ @SeewooLee Ok all is clear, thanks again. If you care to write up an answer, I'll gladly accept. $\endgroup$ – Servaes Nov 15 '18 at 17:09
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As I said, Chebyshev polynomials are exactly what you said.

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