Mahler equations for Zeckendorf numeration
<|reference_start|>Mahler equations for Zeckendorf numeration: 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.<|reference_end|>
@article{carton2024mahler,
    title={Mahler equations for Zeckendorf numeration},
    author={Olivier Carton and Reem Yassawi},
    journal={arXiv preprint arXiv:2405.01953},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01953},
    primaryClass={math.NT cs.FL}
}