Exponential Rationals

Abstract

Several authors studied the so called exponential polynomials, characterized as solutions to the equation $$\begin{aligned} f(x+y) = \sum _{k=1}^nu_i(x)v_i(y). \end{aligned}$$ f ( x + y ) = ∑ k = 1 n u i ( x ) v i ( y ) . In the present paper we deal with a more general equation $$\begin{aligned} \sum _{j=1}^M P_j(x,y) f_j(a_jx + c_j y) = \sum _{k=1}^n u_k(x)v_k(y). \end{aligned}$$ ∑ j = 1 M P j ( x , y ) f j ( a j x + c j y ) = ∑ k = 1 n u k ( x ) v k ( y ) . Here all $$f_j, \ u_k, \ v_k$$ f j , u k , v k are assumed to be unknown scalar functions on $${\mathbb {R}}^d,$$ R d , while $$P_j$$ P j are polynomials. We prove that $$f_j$$ f j are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions.