Log-concavity of Lucas Sequences of first kind

TL;DR

This paper investigates the log-concavity properties of Lucas sequences of the first kind, identifying conditions for their log-concavity and infinite log-concavity, with implications for well-known sequences like Fibonacci and Mersenne.

Contribution

It provides a characterization of when Lucas sequences are log-concave and introduces criteria for their infinite log-concavity, extending understanding of recurrence sequence properties.

Findings

Identifies initial values for log-concavity in Lucas sequences

Determines conditions for infinite log-concavity

Applies results to Fibonacci and Mersenne sequences

Abstract

In these notes we address the study of the log-concave operator acting on Lucas Sequences of first kind. We will find for which initial values a generic Lucas sequence is log-concave, and using this we show when the same sequence is infinite log-concave. The main result will help to find the log-concavity of some well known recurrent sequences like Fibonacci and Mersenne. Some possible generalization for a complete classification of the log-concave operator applied to general linear recurrent sequences is proposed.