On Products of Weierstrass Sigma Functions

Abstract

We prove the following result. Let f : ℂ → ℂ be an even entire function. Assume that there exist ????_j, β_j : ℂ → ℂ with

\( f\left(x+y\right)f\left(x-y\right)=\sum \limits_{\mathrm{j}=1}^4{\alpha}_j(x){\beta}_j(y),\kern0.5em x,y\in \mathbb{C}. \)

Then f(z) = σL(z) · σΛ(z) · e^Az2+C where L and Λ are lattices in ℂ, σ_L is the Weierstrass sigma function associated with the lattice L, and A,C ∈ ℂ.