Fibonacci polynomials

TL;DR

This paper explores Fibonacci polynomials using Heaps of Viennot, providing a combinatorial framework that yields new identities and extends classical orthogonal polynomial theory.

Contribution

It introduces a Heaps-based approach to Fibonacci polynomials, deriving new identities and extending classical orthogonal polynomial theory without restrictions.

Findings

Heaps provide a powerful combinatorial tool for Fibonacci polynomials.

New identities for Fibonacci polynomials are derived using Heaps.

The approach extends classical orthogonal polynomial theory.

Abstract

The Fibonacci polynomials $\big\{F_n(x)\big\}_{n\ge 0}$ have been studied in multiple ways. In this paper we study them by means of the theory of Heaps of Viennot. In this setting our polynomials form a basis $\big\{P_n(x)\big\}_{n\ge 0}$ with $P_n(x)$ monic of degree $n$. This given, we are forced to set $P_n(x)=F_{n+1}(x)$. The Heaps setting extends the Flajolet view of the classical theory of orthogonal polynomials given by a three term recursion. Thus with Heaps most of the identities for our $P_n(x)'s$ can be derived by combinatorial arguments. Using the present setting we derive a variety of new identities. We must mention that the theory of Heaps is presented here without restrictions. This is much more than needed to deal with the Fibonacci polynomials. We do this to convey a flavor of the power of Heaps. In the lecture notes there is a chapter dedicated to Heaps where most of its contents are dedicated to applications of the theory.