Reprint: Arithmetic properties of solutions of a class of functional equations (1929)

Abstract

In this paper, Mahler introduces a method of transcendence now known as Mahler's method. He applies his method to show that the numbers

\( \par \par \sum_{n\ge 0} t(n)\alpha^n\quad\mbox{and}\quad\sum_{n\ge 0}\lfloor n\omega\rfloor \alpha^n \par \par \)

are transcendental for any algebraic number $\alpha$ with $0<|\alpha |<1$ and any positive quadratic irrational number $\omega$, where \( {t(n)\}_{n\ge 0} \) is the Thue-Morse sequence with values in \( {-1,1\} \) and $\rfloor x\lfloor$ denotes the integer part of $x$.

This article is the first in a series of three papers that develops Mahler's method.

Reprint of the author's paper [Math. Ann. 101, 342--366 (1929; JFM 55.0115.01); correction 103, 532 (1930; JFM 56.0185.02)].