On the transcendency of the solutions of a special class of functional equations

Abstract

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and letAssume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the seriesfor fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.