The product $x(t)y(t)$ of two periodic signals with fundamental periods $T_x$ and $T_y$ is not a periodic signal unless $T_x$ and $T_y$ are rational multiples of one another; that is, $T_x = aT_y$ where $a$ is a rational number. Thus, except when such a relationship holds, $x(t)y(t)$ does not have a Fourier series.
When $T_x$ is a rational multiple of $T_y$, the corresponding fundamental frequencies $\omega_x$ and $\omega_y$ are also rational multiples of each other and can be expressed as
$$\omega_x = m\omega_0, ~~ \omega_y = n\omega_0$$ where $m, n$ are positive integers. The product
\begin{align}
x(t)y(t) &= \sum_{i=-\infty}^\infty a_i \exp(ji\omega_x t)\sum_{k=-\infty}^\infty b_i \exp(jk\omega_y t)\\
&= \sum_{i=-\infty}^\infty a_i \exp(jim\omega_0 t)\sum_{k=-\infty}^\infty b_i \exp(jkn\omega_0 t)\\
&= \sum_{i=-\infty}^\infty\sum_{k=-\infty}^\infty a_ib_k \exp(j(im+kn)\omega_0 t)
\end{align}
can be re-arranged into a single sum $\sum_{\ell=-\infty}^\infty c_\ell \exp(j\ell \omega_0 t)$that is recognizable as a Fourier series with fundamental frequency $\omega_0$. Many of the Fourier coefficients $c_\ell$ in this series have value $0$; in particular, $c_1 = 0$.
