On rational addition theorems ✩

We investigate the functions which admit an addition theorem of the form f (t + s) = n i=1 yi (t)ui (s) m j=1 zj (t)vj (s) . Our approach is based on the reduction to a system of differential equations. (All functions are supposed to be continuously differentiable on some interval.) The concepts of joint linear dependence and joint quadratic dependence of two families of functions are introduced. It’s proved that there are only two possibilities in the case of jointly linearly independent {ui } and {vj }: (a) the function f is a ratio of quasi-polynomials, (b) the families {yi } and {zj } are jointly quadratically dependent. The second possibility is studied for m = n = 2. We apply our results to the solving of the functional equation f (t + s) = y1(t)y2(s) −y2(t)y1(s) z1(t)z2(s) −z2(t)z1(s) . This equation and its special cases have important applications in mathematical physics and were studied by several authors with the assumption of analyticity of the solutions. We show that the class of solutions remains the same if one looks for the differentiable solutions on an interval.  2003 Elsevier Science (USA). All rights reserved