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}

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 $T_3$?

• dunno i would call it a complete answer. Apr 13, 2018 at 8:09
• I have every confidence that you could figure it out without my hints.
– user28715
Apr 13, 2018 at 8:12
• Just for clarity: homework or self-study questions are within the scope of this site, but they must include enough of the working to show where the OP loses understanding. You are correct that actually solving the homework is outside the scope.
– Peter K.
Apr 13, 2018 at 12:43