Fourier Series of Aperiodic convolution of periodic functions
okay, now that the popcorn is eaten and my fingers are clean...

$$ x(t) = \sum\limits_{m=-\infty}^{+\infty} c_m^x e^{j 2 \pi (m/T) t} $$

$$ y(t) = \sum\limits_{m=-\infty}^{+\infty} c_m^y e^{j 2 \pi (m/T) t} $$

then

$$\begin{align}
 z(t) &amp;= \sum\limits_{m=-\infty}^{+\infty} c_m^z e^{j 2 \pi (m/T) t} \\
\\
  &amp;= x(t) \circledast y(t) \\
\\
  &amp;= \int\limits_{-\infty}^{+\infty} x(\tau) y(t-\tau) \, d\tau \\
\\
  &amp;= \int\limits_{-\infty}^{+\infty} \sum\limits_{m=-\infty}^{+\infty} c_m^x e^{j 2 \pi (m/T) \tau} \sum\limits_{n=-\infty}^{+\infty} c_n^y e^{j 2 \pi (n/T) (t-\tau)}  \, d\tau \\
\\
  &amp;= \sum\limits_{n=-\infty}^{+\infty} \sum\limits_{m=-\infty}^{+\infty} c_m^x c_n^y  \int\limits_{-\infty}^{+\infty} e^{j 2 \pi (m/T) \tau} e^{j 2 \pi (n/T) (t-\tau)}  \, d\tau \\
\\
  &amp;= \sum\limits_{n=-\infty}^{+\infty} \sum\limits_{m=-\infty}^{+\infty} c_m^x c_n^y \int\limits_{-\infty}^{+\infty} e^{j 2 \pi ((m-n)/T) \tau} \, d\tau \ e^{j 2 \pi (n/T) t} \\
\end{align} $$

this integral: $\int\limits_{-\infty}^{+\infty} e^{j 2 \pi ((m-n)/T) \tau} \, d\tau$ ain't converging when $n \ne m$ and certainly not when $n=m$.

you will not be getting finite values for $c_m^z$.