On the connection between hyperelliptic systems of sequences and functions

We study the connection between $1$-periodic solutions of the functional equation
$$ f(x+y)g(x-y)=\sum_{j=1}^N\varphi_j(x)\psi_j(y) \quad (x,y\in \mathbb C) $$
and some sequences of special kind. As an application we solve the equation in the case when $g$ is Jacobi theta function.