2019/12/23 by Edward E. Allen, Allen, Edward E., Katherine Riley +3
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1912.10943
openalex publication_date 2019/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Lucas polynomials are polynomials in s1 and s2 defined recursively by \0\=0, \1\=1, and \m\=s1\m-1\+s2\m-2\ for m ≥ 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 s1, sr, and s2r. 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.