Zeros and Orthogonality of generalized Fibonacci polynomials

TL;DR

The paper addresses when second-order GFP sequences are orthogonal and provides an explicit weight measure in a targeted Favard-case, enabling a concrete classification of familiar GFP families into orthogonal or non-orthogonal groups. It introduces a root-finding method based on GFP Binet formulas and translates orthogonality into a probabilistic context via the Karlin–McGregor representation, linking GFPs to birth-and-death Markov processes. The authors identify which GFP families can drive discrete- and continuous-time random walks (notably Lucas-type GFPs) and provide explicit coefficient conditions yielding stochastic transition structures, with illustrative examples from Chebyshev, Fermat-Lucas, and Morgan-Voyce polynomials. These results deepen the bridge between orthogonal polynomials and stochastic processes, suggesting new avenues for modeling and analysis in both communities.

Abstract

This paper analyzes the concept of orthogonality in second-order polynomial sequences that have Binet formula similar to that of the Fibonacci and Lucas numbers, referred to as Generalized Fibonacci Polynomials (GFP). We give a technique to find roots of the GFP. As a corollary of this result, we give an alternative proof of a special case of Favard's Theorem. The general case of Favard's Theorem guarantees that there is a measure to determine whether a sequence of second-order polynomials is orthogonal or not. However, the theorem does not provide an explicit such measure. Our special case gives both the explicit measure and the relationship between the second-order recurrence and orthogonality, demonstrating whether the GFP polynomials are orthogonal or not. This allows us to classify which of familiar GFPs are orthogonal and which are not. Some familiar orthogonal polynomials include the Fermat, Fermat-Lucas, both types of Chebyshev polynomials, both types of Morgan-Voyce polynomials, and Vieta and Vieta-Lucas polynomials. However, we prove that the Fibonacci, Lucas, Pell, and Pell-Lucas sequences are not orthogonal. In Section \ref{sectionrw}, we give a brief description of discrete--time and continuous--time Morkov chains with special emphasis on birth-and-death stochastic processes. We find sufficient conditions on the polynomial's coefficients under which a given family of orthogonal polynomial induces a Markov chain. These families of orthogonal polynomials include Chebyshev polynomials of first kind and Fermat-Lucas. In the final section, we highlight some connections between orthogonal polynomials and Markov processes. These relations are not new but seem to have been somewhat forgotten. We do so to draw the attention of researchers in the orthogonal polynomial and probability communities for further collaboration.

Figures (2)

• Figure 1: Transitions between states.
• Figure 2: Transitions between states.

Theorems & Definitions (29)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1: FlorezJC
• Lemma 3.2: FlorezJC
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Theorem 3.5
• Proposition 3.6