Identically vanishing $k$-generalized Fibonacci polynomials

TL;DR

This paper investigates the properties of $k$-Fibonacci polynomials for negative indices, revealing when they vanish identically, and explores their roots, coefficients, and connections to the Skolem problem, extending existing theory.

Contribution

It identifies indices where $k$-Fibonacci polynomials vanish for negative $n$, derives their coefficients, roots, and bounds, and extends the $k$-nomial triangle to negative indices.

Findings

Identifies indices where polynomials vanish for negative $n$

Provides explicit formulas for polynomial coefficients and roots

Extends the $k$-nomial triangle to negative indices

Abstract

The recurrence for the $k$-Fibonacci polynomials is usually iterated upwards to positive values of $n$ only. When the recurrence is iterated downwards to $n<0$, there are indices where the polynomials vanish identically. This fact does not seem to have been noted in the literature. We derive the set of such indices. We establish the connection of our results to the solution of the Skolem problem for the $k$-Fibonacci numbers. For $k\ge3$ and $n<0$, we show that the degree of the polynomial does not increase monotonically with $|n|$. The so-called `left-justified $k$-nomial triangle' is extended to treat negative indices. We derive expressions for the individual polynomial coefficients (the elementary symmetric polynomials of the roots). We present results for the properties of the polynomials, for both $n>0$ and $n<0$, including factorization of the polynomials and properties of the roots. Results are also derived for real roots. We present new, tighter, bounds on the amplitudes of the nonzero roots. We derive new combinatorial sums for the polynomial coefficients, which are more concise and computationally efficient than previously published expressions.