Popken, Jan; Mahler, Kurt

Reprint: Ein neues Prinzip für Transzendenzbeweise (1935)

Doc. Math. Extra Vol. Mahler Selecta, 449-457 (2019)
DOI: 10.25537/dm.2019.SB-449-457

Summary

In this paper, Popken and Mahler extend a result in \textit{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. \par Reprint of the authors' paper [Proc. Akad. Wet. Amsterdam 38, 864--871 (1935; Zbl 0012.34101; JFM 61.0187.02)].

Mathematics Subject Classification

11-03, 11J81

Affiliation

Popken, Jan
Email:
Mahler, Kurt
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