Leonhard Euler and a q-analogue of the logarithm

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

2007-09-12

arxiv.org/abs/0709.1788v1

Proc AMS 137 (2009), no. 5, 1663-1676

Mathematics

Classical Analysis and ODEs

13 pages, to appear in Proc. AMS

Scientific paper

10.1090/S0002-9939-08-09374-X

We study a q-logarithm which was introduced by Euler and give some of its properties. This q-logarithm did not get much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751-paper and 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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