Logarithm of Irrationals and Beatty Sequences

TL;DR

This paper establishes identities connecting the logarithms of irrational numbers with series involving Beatty sequence ratios and explores their relation to Sturmian sequences, revealing new mathematical representations and connections.

Contribution

It introduces novel identities linking irrational logarithms to Beatty sequence ratios and characterizes Sturmian sequences through these ratios.

Findings

Derived a series representation for logarithms of irrationals using Beatty sequences

Connected Beatty sequence ratios to Sturmian sequences

Identified a new identity resembling Frullani's Integral in a discrete setting

Abstract

In this paper we find an identity that gives a representation for the logarithm of any two irrational numbers $a, b >1$ in terms of a series whose terms are ratios of elements from the Beatty Sequences generated by these two numbers. We also show that Sturmian sequences can be defined in terms of these ratios. Furthermore, we find an identity for such series that bears a superficial resemblance to (a discrete version of) Frullani's Integral.