Functional equations and Weierstrass sigma-functions

Abstract

It is proved that if an entire function f: ℂ → ℂ satisfies an equation of the form α 1(x)β 1(y) + α 2(x)β 2(y) + α 3(x)β 3(y), x,y ∈ C, for some α j , β j : ℂ → ℂ and there exist no \({\widetilde \alpha _j}\) and ˜\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az 2 + Bz + C) ∙ σ Γ(z - z 1) ∙ σ Γ(z - z 2), where Γ is a lattice in ℂ; σ Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z 1, z 2 ∈ ℂ; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).