How to do convolution in Fourier Series?
Following your notation, if we define the coefficients of the Fourier series of $z(t)=x(t)y(t)$ as $c_k$:

$$c_k=\sum_{l=-\infty}^{\infty}a_lb_{k-l}$$

Notice that the $k$ in which you evaluate the left side of the equality has to be the same as the $k$ in the summation. Therefore, for example, if you want to find $c_0$, you should calculate:

$$c_0=\sum_{l=-\infty}^{\infty}a_lb_{-l}$$

Then you have to calculate that summation. Given that $a_l$ and $b_{-l}$ are non-zero only if $l$ equals $1$ or $-1$:

$$\begin{align}
c_0
&amp;=\sum_{l=-\infty}^{\infty}a_lb_{-l} \\
&amp;=a_{-1}b_1 + a_1b_{-1} \\
&amp;=\frac12 \frac{1}{2j} - \frac12 \frac{1}{2j}=0
\end{align}$$

I think this might have clarified the notation for you. You can go on and calculate the rest of the coefficients $c_k$ for all values of $k$.