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}} &amp; \hspace{3mm} \text{even}\;k \\
                a_k e^{-7jk\frac{2\pi}{T}} &amp; \hspace{3mm} \text{odd}\;k \\
                \end{array} \right.
\end{aligned}
Since this looks like a homework and solving people's homework is outside the scope of this group, here are some hints

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 $T_3$?