SOME GENERALIZED LACUNARY POWER SERIES WITH ALGEBRAIC COEFFICIENTS FOR MAHLER'S $U-$NUMBER ARGUMENTS

Abstract

In this work, we show that under certain conditions the values of some generalized lacunary power series with algebraic coefficients for Mahler's $U_{m}-$number arguments belong to either a certain algebraic number field or $\bigcup_{i=1}^{t} U_{i}$ in Mahler's classification of the complex numbers, where $t$ denotes a positive rational integer dependent on the coefficients of the given series and on the argument. Moreover, the obtained results are adapted to the field $\mathbb{Q}_{p}$ of $p-$adic numbers.