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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Leonhard Euler and a $q$-analogue of the logarithm
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by Erik Koelink and Walter Van Assche PDF
Proc. Amer. Math. Soc. 137 (2009), 1663-1676 Request permission


We study a $q$-logarithm which was introduced by Euler and give some of its properties. This $q$-logarithm has not received much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751 paper and in a 1734 letter to Daniel Bernoulli. The corresponding $q$-analogue of the dilogarithm is introduced. The relation to the values at $1$ and $2$ of a $q$-analogue of the zeta function is given. We briefly describe some other $q$-logarithms that have appeared in the recent literature.
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Additional Information
  • Erik Koelink
  • Affiliation: IMAPP, FNWI, Radboud Universiteit, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
  • Email:
  • Walter Van Assche
  • Affiliation: Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
  • MR Author ID: 176825
  • ORCID: 0000-0003-3446-6936
  • Email:
  • Received by editor(s): March 6, 2007
  • Published electronically: December 12, 2008
  • Additional Notes: The second author was supported by research grant OT/04/21 of Katholieke Universiteit Leuven, research project G.0455.04 of FWO-Vlaanderen, and INTAS research network 03-51-6637

  • Dedicated: On the 300th anniversary of Euler’s birth
  • Communicated by: Peter A. Clarkson
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1663-1676
  • MSC (2000): Primary 33B30, 33E30
  • DOI:
  • MathSciNet review: 2470825