Solution of functional equations related to elliptic functions (Q1661309)

The author solves the functional equations \[ f(x+y)g(x-y)=\sum_{j=1,2,3}\alpha_j(x)\beta_j(y) \] and \[ f_1(x+z)f_2(y+z)f_3(x+y-z)=\sum_{j=1,2,3,4}\phi_j(x,y)\psi_j(z). \] All the functions are unknown and entire. These equations are connected with addition theorems for elliptic functions, as for instance, the relation \[ \sigma(x + y)\sigma(x- y) = \sigma^2(x)\sigma^2(y)(\wp(y) -\wp(x)). \] The author proves that the solutions of the functional equations are elliptic functions and quasi-polynomials \(\sum_{j=1}^s P_j(z) \exp (\lambda_j z)\), where \( P_j(z)\) are polynomials.