On a Multilinear Functional Equation

Abstract

The following functional equation is solved: $$f\left( {{x_1} + z} \right) \cdots f\left( {{x_2} + z} \right)f\left( {{x_1} + \cdots + {x_{s - 1}} - z} \right) = {\phi _1}\left( x \right){\psi _1}\left( z \right) + \cdots + {\phi _m}\left( x \right){\psi _m}\left( z \right),$$ where x =(x1,…,xs−1), for the unknowns $$f,{\psi _j}:\mathbb{C} \to \mathbb{C}$$ and $${\phi _j}:{\mathbb{C}^{s - 1}} \to \mathbb{C}$$ for s ≥ 3 and m ≤ 4s − 5.