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Leonhard Euler and a $q$-analogue of the logarithm

Authors: Erik Koelink and Walter Van Assche
Journal: Proc. Amer. Math. Soc. 137 (2009), 1663-1676
MSC (2000): Primary 33B30, 33E30
Published electronically: December 12, 2008
MathSciNet review: 2470825
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Abstract: 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

Walter Van Assche
Affiliation: Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
MR Author ID: 176825
ORCID: 0000-0003-3446-6936

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
Article copyright: © Copyright 2008 American Mathematical Society