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 $$\sum_{n\ge 1}\frac{q^{2n}}{(1-q^{2n})^2},\quad \sum_{n\ge 1}\frac{q^{2n}}{(1-q^{2n})^4},\quad \sum_{n\ge 1}\frac{q^{2n}}{(1-q^{2n})^6}$$ is a transcendental number.

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

Reprint: Ein neues Prinzip für Transzendenzbeweise (1935)

Jan Popken

Kurt Mahler

Abstract