Mahler equations for Zeckendorf numeration
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.01953
- arXiv:
- arXiv:2405.01953
- Bibcode:
- 2024arXiv240501953C
- Keywords:
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- Mathematics - Number Theory;
- Computer Science - Formal Languages and Automata Theory;
- 11B85;
- 68Q45
- E-Print:
- 33 pages, 6 figures