Calculation of hyperelliptic systems of sequences of rank 4

Formulas for sequences of complex numbers satisfying functional relations of bilinear type are investigated. The results obtained are used in describing all 1-periodic entire functions $f,g\colon \mathbb{C}\to\mathbb{C}$ satisfying $f(x+y)g(x-y)=\phi_1(x)\psi_1(y)+\dots+\phi_4(x)\psi_4(y)$ for some $\phi_j,\psi_j\colon \mathbb{C}\to\mathbb{C}$.