Multivariable Lucas Polynomials and Lucanomials

TL;DR

This paper extends Lucas polynomials from two variables to multiple variables, providing recursive definitions, polynomial properties, combinatorial interpretations, and generating functions for the multivariable case.

Contribution

The paper introduces a multivariable generalization of Lucas polynomials, including recursive definitions, polynomial binomial analogues, and combinatorial interpretations.

Findings

Multivariable Lucas polynomials are well-defined and polynomial.

Binomial analogues of multivariable Lucas polynomials are polynomial.

Recursive generating functions for multivariable Lucas polynomials are established.

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.