Hyperelliptic systems of sequences of rank 4

Sequences of complex numbers satisfying functional relations of bilinear type are investigated. The results obtained are used in describing all 1-periodic entire functions $f\colon \mathbb C\to\mathbb C$ such that the expansion ${f(x+y)f(x-y)}=\varphi_1(x)\psi_1(y)+\dots+\varphi_4(x)\psi_4(y)$ holds for some $\varphi_j,\psi_j\colon\mathbb C\to\mathbb C$.
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