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

Source abstract

Let a ( z ) and b ( w ) be two rational functions in z or W with algebraic coefficients, where a (0) = 0 and let Assume 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 series for fixed w is a transcendental function of z . Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.