Sign-alternating Gibonacci polynomials

TL;DR

This paper studies sign-alternating Gibonacci polynomials, revealing their real, distinct roots, explicit bounds, and closed-form expressions, with applications to combinatorial games and lattice structures.

Contribution

It introduces new properties of sign-alternating Gibonacci polynomials, including root geometry, explicit bounds, and closed-form formulas, and applies these to combinatorial and algebraic structures.

Findings

Roots are real and distinct with explicit bounds.

Derived Binet-type closed-form expressions.

Applied results to combinatorial game and lattice enumeration.

Abstract

We consider various properties and manifestations of some sign-alternating univariate polynomials borne of right-triangular integer arrays related to certain generalizations of the Fibonacci sequence. Using a theory of the root geometry of polynomial sequences developed by J. L. Gross, T. Mansour, T. W. Tucker, and D. G. L. Wang, we show that the roots of these `sign-alternating Gibonacci polynomials' are real and distinct, and we obtain explicit bounds on these roots. We also derive Binet-type closed expressions for the polynomials. Some of these results are applied to resolve finiteness questions pertaining to a one-player combinatorial game (or puzzle) modelled after a well-known puzzle we call the `Networked-numbers Game.' Elsewhere, the first- and second-named authors, in collaboration with A. Nance, have found rank symmetric `diamond-colored' distributive lattices naturally related to certain representations of the special linear Lie algebras. Those lattice cardinalities can be computed using sign-alternating Fibonacci polynomials, and the lattice rank generating functions correspond to the rows of some new and easily defined triangular integer arrays. Here, we present Gibonaccian, and in particular Lucasian, versions of those symmetric Fibonaccian lattices/results, but without the algebraic context of the latter.