On certain generalizations of the Levi-Civita and Wilson functional   equations
  We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials. 
J. M. Almira and E. V. Shulman
math.CA