On Mahler's S-numbers, T-numbers, and U-numbers

Abstract

In this work, we consider some power series with algebraic coefficients from a certain algebraic number field, whose radiuses of convergence are infinite. We show that under certain conditions these series take transcendental values at non-zero algebraic number arguments, and we determine the classes to which these transcendental values belong in Mahler's classification. Then we consider these series for certain Liouville number arguments and obtain similar results.