The characterization of theta functions by functional equations
The author solves the functional equation \[ \theta (u+v) \theta (u-v)=f_1 (u) g_1 (v)+f_2 (u) g_2 (v) \tag{1} \] which occurs in the theory of theta functions. The main result is the following theorem. Suppose \(\theta: \mathbb{R}^n\to \mathbb{C}\) is continuous and assume that there are functions \(f_1, f_2, g_1, g_2: \mathbb{R}^n\to \mathbb{C}\) such that (1) is valid for all \(u,v\in \mathbb{R}^n\). If \(f_1\), \(f_2\) or \(g_1\), \(g_2\) are linearly independent over \(\mathbb{C}\), then \[ \theta (u)=c \exp (Q(U)+L(u)), \qquad u\in \mathbb{R}^n. \] If both \(f_1\), \(f_2\) and \(g_1\), \(g_2\) are linearly independent, then \[ \theta (u)=c\exp (Q(U)+L(u))\cdot B(u), \qquad u\in \mathbb{R}^n, \] where \(c\in \mathbb{C}\) and the function \(B(u)\) is an elliptic function, or a sinus function, or a linear function of \(u\). A generalization of the equation (1) is the equation \[ \theta (u+v)\theta (u-v)=\sum^m_{i=1} f_i (u) g_i (v), \] satisfied by the theta functions of several complex variables [cf. \textit{M. Bonk}, Math. Ann. 298, No. 4, 591-610 (1994; Zbl 0791.39009)].
scientific article; zbMATH DE number 878363
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