Multivariate Fibonacci-like Polynomials and their Applications

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.