Multivariate Fibonacci-like Polynomials and their Applications

TL;DR

This paper introduces multivariate Fibonacci-like polynomials, generalizes classical properties, and explores their algebraic and combinatorial connections, expanding the theoretical understanding of Fibonacci polynomials.

Contribution

It extends Fibonacci polynomials to multiple variables, provides explicit formulas, proves irreducibility, and links to various combinatorial structures.

Findings

Generalized Fibonacci polynomials to multiple variables.

Derived explicit formulas and generating functions.

Proved irreducibility over complex numbers for certain cases.

Abstract

The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend several well-known results such as the explicit Binet formula and a Cassini-like identity, and use these to prove that the $r$-Fibonacci polynomials are irreducible over $\mathbb{C}$ for $n \geq r \geq 3$. Additionally, we derive an explicit sum formula and a generalized generating function. Using these results, we establish connections to ordinary Bell polynomials, exponential Bell polynomials, Fubini numbers, and integer and set partitions.