On the rank of odd hyper-quasi-polynomials

Abstract

Given any nonzero entire function g: ℂ → ℂ, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+∙∙∙+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ℂ → ℂ. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.