Multivariate Fibonacci-like Polynomials and their Applications

The Fibonacci polynomials are defined recursively as $f_n(x)=x{f}_{n-1}(x)+f_{n-2}(x)$, 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 $n\ge r\ge 3$. 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.