## Reprint: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen (1929)

##### Doc. Math. Extra Vol. Mahler Selecta, 191-216 (2019)
DOI: 10.25537/10.25537/dm.2019.SB-191-216

### Summary

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 $\par \par \sum_{n\ge 0} t(n)\alpha^n\quad\mbox{and}\quad\sum_{n\ge 0}\lfloor n\omega\rfloor \alpha^n \par \par$ \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 $x$. \par This article is the first in a series of three papers that develops Mahler's method. \par Reprint of the author's paper [Math. Ann. 101, 342--366 (1929; JFM 55.0115.01); correction 103, 532 (1930; JFM 56.0185.02)].

11-03, 11J81

Mahler's method