Leonhard Euler and a q-analogue of the logarithm

TL;DR

This paper explores Euler's q-logarithm, deriving its fundamental properties, introducing a q-analogue of the dilogarithm, and relating these to q-analogues of the zeta function, highlighting its historical and mathematical significance.

Contribution

It provides a comprehensive analysis of Euler's q-logarithm, including properties, a new q-analogue of the dilogarithm, and connections to q-zeta functions, which were previously underexplored.

Findings

Derived basic properties of Euler's q-logarithm.

Introduced a q-analogue of the dilogarithm.

Connected q-logarithm values to q-zeta functions.

Abstract

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.