Fibonacci Sequence, Recurrence Relations, Discrete Probability Distributions and Linear Convolution

TL;DR

This paper investigates the properties of Fibonacci and related linear recurrence sequences through the lens of probability distributions and linear convolution, revealing new limiting behaviors and the influence of characteristic roots.

Contribution

It generalizes known properties of Fibonacci-based probability distributions and analyzes the impact of the dominant root on sequence maxima after convolution.

Findings

Generalized limiting properties of probability distributions from recurrence sequences.

Identified optimal properties of Fibonacci sequences in probabilistic context.

Showed how the largest root influences the maximum location in convolved sequences.

Abstract

The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer coefficients from the point of view of probability distributions that they induce. We obtain the generalizations of some of the known limiting properties of these probability distributions and present certain optimal properties of the classical Fibonacci sequence in this context. In addition, we also look at the self linear convolution of linear recurrence relations with positive integer coefficients. Analysis of self linear convolution is focused towards locating the maximum in the resulting sequence. This analysis, also highlights the influence that the largest positive real root, of the "characteristic equation" of the linear recurrence relations with positive integer coefficients, has on the location of the maximum. In particular, when the largest positive real root is 2,the location of the maximum is shown to depend on whether the sequence length is odd or even.