Multivariable Lucas Polynomials and Lucanomials

Abstract

Lucas polynomials are polynomials in $s_1$ and $s_2$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{m\}=s_1\{m-1\}+s_2\{m-2\}$ for $m \geq 2$. We generalize Lucas polynomials from 2-variable polynomials to multivariable polynomials. This is done by first defining $r$-Lucas polynomials $\{m\}_r$ in the variables $s_1$, $s_r$, and $s_{2r}$. We show that the binomial analogues of the $r$-Lucas polynomials are polynomial and give a combinatorial interpretation for them. We then extend the generalization of Lucas polynomials to an arbitrarily large set of variables. Recursively defined generating functions are given for these multivariable Lucas polynomials. We conclude by giving additional applications and insights.