Hyperquasipolynomials for the theta-function
An entire function \(f\) is hyperquasipolynomial w.r.t. a function \(g: \mathbb{C}\to \mathbb{C}\) if \(f\) satisfies \[ f(x+y)g(x-y)=\sum_{j=1}^r \alpha_j(x) \beta_j(y) \] for some \(r\in \mathbb{N}\) and functions \(\alpha_j, \beta_j: \mathbb{C} \to \mathbb{C}\) [\textit{V. A. Bykovskii}, Funct. Anal. Appl. 50, No. 3, 193--203 (2016); translation from Funkts. Anal. Prilozh. 50, No. 3, 34--46 (2016; Zbl 1360.30023)]. The set of hyperquasipolynomials w.r.t. \(g\) is denoted by \(\mathcal{F}(g)\). For \(h: \mathbb{C}\to \mathbb{C} \), by \(V(h)\) we denote the linear span of the set of functions of the form \(z\mapsto Q(z) h^{(k)}(z+z_0)\), where \(k\in \mathbb{N}\cup\{0\}\), \(z_0\in \mathbb{C}\), and \(Q\) is a quasipolynomial. Theorem. If \(g\in V(\theta)\setminus \{0\}\), then \(\mathcal F(g)=V(\theta)\), where \(\theta\) is the Jacobi theta-function.
scientific article; zbMATH DE number 7000563