Divisors and specializations of Lucas polynomials

TL;DR

This paper explores the divisibility and symmetry properties of Lucas polynomials, including their specializations to Pell and Delannoy numbers, and introduces a structural decomposition into flat and sharp analogs.

Contribution

It presents a novel structural decomposition of Lucas polynomials into flat and sharp forms, advancing understanding of their divisibility and symmetry properties.

Findings

Decomposition of Lucas polynomials into flat and sharp analogs

Specializations to Pell and Delannoy numbers analyzed

Insights into divisibility and symmetry properties

Abstract

Three-term recurrences have infused stupendous amount of research in a broad spectrum of the sciences, such as orthogonal polynomials (in special functions) and lattice paths (in enumerative combinatorics). Among these are the Lucas polynomials, which have seen a recent true revival. In this paper one of the themes of investigation is the specialization to the Pell and Delannoy numbers. The underpinning motivation comprises primarily of divisibility and symmetry. One of the most remarkable findings is a structural decomposition of the Lucas polynomials into what we term as flat and sharp analogs.