2023/09/15 by Park, Sejin, Phillips, Etienne, Qi, Peikai +2
#11B39 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2309.08123
The Fibonacci polynomials are defined recursively as fn(x)=xfn-1(x)+fn-2(x), where f0(x) = 0 and f1(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 ℂ for n ≥ r ≥ 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.