# Fourier coefficients of sum of two functions with different fundamental periods?

If we assume $\quad x(t)\leftrightarrow a_k\:$ and it is periodic with fundamental period T.
How can we determine the fourier coefficients of the sum

$x(t-7)+x(-2t+3)$

I know that $x(t-7)\leftrightarrow e^{-jk7\frac{2\pi}{T}}a_k$

But, since these two functions have different fundamental periods we cannot apply linear combination rule. So how can we find the related coefficients? Thank you.

EDIT: After Stanley Pawlukiewicz's hints

$x(t-7)=\sum\limits_{k=-\infty}^{\infty}a_ke^{jk\dfrac{2\pi t}{T}}e^{-7jk\dfrac{2\pi}{T}}$

$x(-2t+3)=\sum\limits_{k=-\infty}^{\infty}a_{-k}e^{jk\dfrac{4\pi t}{T}}e^{-3jk\dfrac{4\pi}{T}}$

This new function is periodic with $\:T_0=T\:$ since $\:LCM(T,T/2)=T$

Thus,\begin{aligned} \bar{a_k} = \left\{ \begin{array}{cc} a_k e^{-7jk\frac{2\pi}{T}}+a_{\frac{-k}{2}}e^{-3jk\frac{2\pi }{T}} & \hspace{3mm} \text{even}\;k \\ a_k e^{-7jk\frac{2\pi}{T}} & \hspace{3mm} \text{odd}\;k \\ \end{array} \right. \end{aligned}

To have a Fourier Series, a function needs to be periodic. $$g(t)=g(t+T_1) \quad \Rightarrow \sum_k \left(a_k \sin(2\pi k \frac{t}{T_1}) + b_k \cos (2\pi k \frac{t}{T_1}) \right)$$ and $$h(t)=h(t+T_2) \quad \Rightarrow \sum_k \left(c_k \sin(2\pi k \frac{t}{T_2}) + d_k \cos (2\pi k \frac{t}{T_2}) \right)$$ If $z(t)=g(t)+h(t)$
Under what circumstances of $T_1$ and $T_2$ would $$z(t)=z(t+T_3) \quad \text{?}$$ an what circumstances of $T_1$ and $T_2$ would $$z(t)\ne z(t+T_3) \quad \text{for any} \quad T_3$$
and if there were circumstance where $z(t)=z(t+T_3)$,
Could we re express $g(t)$ and $h(t)$ as Fourier Series periodic in $T_3$?