Mahler equations for Zeckendorf numeration

TL;DR

This paper introduces generalized Z-Mahler equations linked to Zeckendorf numeration, establishing a correspondence with Z-regular sequences and automata, thereby extending previous theoretical results.

Contribution

It generalizes existing results on Mahler equations and Zeckendorf numeration, providing new characterizations and automata constructions for Z-regular sequences.

Findings

Z-regular sequences correspond to solutions of Z-Mahler equations

Isolating Z-Mahler equations characterize Z-regular sequences

New automata construction for classical q-regular sequences

Abstract

We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system. We show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a Z-Mahler equation. Conversely, if the Z-Mahler equation is isolating, then its solutions define Z-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating Z-Mahler equations whose solutions do not define Z-regular sequences. Our proof yields a new construction of weighted automata that generate classical q-regular sequences.