Relations de r\'ecurrence lin\'eaires, primitivit\'e et loi de Benford

TL;DR

This paper demonstrates that many linear recurrence sequences with non-negative coefficients follow Benford's Law in various bases, using Perron-Frobenius theory applied to the sequence's characteristic polynomial.

Contribution

It establishes a connection between linear recurrence sequences and Benford's Law, providing a theoretical proof based on Perron-Frobenius theory.

Findings

Sequences defined by linear difference equations follow Benford's Law in multiple bases.

The proof utilizes Perron-Frobenius theory and the companion matrix of the characteristic polynomial.

Many such sequences exhibit Benford behavior under suitable conditions.

Abstract

We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in many bases $b>2$. Our techniques rely on Perron-Frobenius theory via the companion matrix of the characteristic polynomial of the defining equation.