Reprint: Ein neues Prinzip für Transzendenzbeweise (1935)

Abstract

In this paper, Popken and Mahler extend a result in J. Popken's dissertation [Über arithmetische Eigenschaften analytischer Funktionen (German). Groningen: Univ. Groningen (Diss.) (1935; Zbl 0013.27004; JFM 61.1136.01)]. In particular, they show that, for any $q$ with $0<|q|<1$, at least one of the three numbers ${\sum }_{n\ge 1}\frac{q^{2n}}{(1-q^{2n}{)}^2},{\sum }_{n\ge 1}\frac{q^{2n}}{(1-q^{2n}{)}^4},{\sum }_{n\ge 1}\frac{q^{2n}}{(1-q^{2n}{)}^6}$ is a transcendental number.

Reprint of the authors' paper [Proc. Akad. Wet. Amsterdam 38, 864--871 (1935; Zbl 0012.34101; JFM 61.0187.02)].