Resolution of the Skolem Problem for $k$-Generalized Lucas Sequences

TL;DR

This paper completely characterizes the zero-distribution of $k$-generalized Lucas sequences, especially at negative indices, and determines the exact zero-multiplicity for all $k$, solving Skolem's problem in this context.

Contribution

It provides a complete solution to Skolem's problem for $k$-generalized Lucas sequences, including zero-distribution and zero-multiplicity characterization.

Findings

Zero-multiplicity $oxed{rac{(k-1)(k-2)}{2}}$ for all $k$

Characterization of all negative indices where the sequence is zero

Complete solution to Skolem's problem for these sequences

Abstract

This paper provides a complete solution to Skolem's problem for the $k$-generalized Lucas sequence $(L_n^{(k)})_{n \in \mathbb{Z}}$ with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this sequence by identifying and bounding all indices $n < 0$ such that $L_n^{(k)} = 0$. Our central result establishes that the zero-multiplicity $\delta_k$ of the sequence is $(k-1)(k-2)/2$ for all $k.$