Two basic questions related to complex Fourier series
First, or second, with complex notation and the binomial theorem,

$$\sin^5 x=\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^5 = \frac{e^{i5x}-5e^{i3x}+10e^{ix}-10e^{-ix}+5e^{-3ix}-e^{-i5x}}{32i} = \frac{\sin(5x)-5\sin(3x)+10\sin x}{16}$$

On the other hand,

\begin{align}
\sin(5x) &amp;= Im(e^{i5x}) = Im([\cos x+i\sin x]^5) \\
&amp;= 5\cos^4x\sin x - 10\cos^2x\sin^3x + \sin^5x\\
&amp;= 5(1-2\sin^2 x+\sin^4 x)\sin x-10(1-\sin^2x)\sin^3x+ \sin^5x\\
&amp;= 5\sin x - 20\sin^3x +16\sin ^5x
\end{align}

so you got the coefficients right, but the notation reversed.



In the second task, is the star connected to convolution, cepstrum transforms or similar?

No, it is just $g(x)=\exp(α x)=e^{α x}$ defined on the interval $[-\pi, \pi]$. Then 

\begin{align}
c_k
&amp;=\frac1{2\pi}\int_{-\pi}^\pi g(x)e^{-ikx}dx
 =\frac1{2\pi}\int_{-\pi}^\pi e^{(α-ik)x}dx
\\
&amp;=\frac1{2\pi}\left[\frac{e^{(α-ik)x}}{α-ik}\right]_{x=-\pi}^{x=\pi}\\
\end{align}