Given a bounded real function $f(x)$ with period P, we can express it as Fourier series:
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty\left[a_n\cos \left( \frac{2n\pi}{P}x \right) + b_n\sin \left( \frac{2n\pi}{P}x \right) \right] $$
Now we can express it as a formal Taylor series:
$$ f(x) = \frac{a_0}{2} + \sum_{k=0}^\infty \left(\alpha_k x^{2k} + \beta_k x^{2k+1}\right)$$
where the coefficients $\alpha_k$ and $\beta_k$ arise from the Taylor developments of $\cos$ and $\sin$:
$$ \alpha_k = \frac{(-1)^k}{(2k)!} \sum_{n=1}^\infty a_n\left(\frac{2n\pi}{P}\right)^{2k}, \text{ and } \beta_k = \frac{(-1)^k}{(2k+1)!} \sum_{n=1}^\infty b_n\left(\frac{2n\pi}{P}\right)^{2k+1} \quad (*)$$
So my question is, for every real periodic and analytic function:
$$f(x) = \frac{a_0}{2} + \sum_{k=0}^\infty \gamma_k x^k $$
is it is true that we can find sequences $(a_n)$ and $(b_n)$ satisfaying $\gamma_{2k} = \alpha_k(a_n)$ and $\gamma_{2k+1} = \beta_k(b_n)$?