Mahler equations for Zeckendorf numeration
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917342645059584 |
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| author | Carton, Olivier Yassawi, Reem |
| author_facet | Carton, Olivier Yassawi, Reem |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01953 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mahler equations for Zeckendorf numeration Carton, Olivier Yassawi, Reem Number Theory Formal Languages and Automata Theory 11B85, 68Q45 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. |
| title | Mahler equations for Zeckendorf numeration |
| topic | Number Theory Formal Languages and Automata Theory 11B85, 68Q45 |
| url | https://arxiv.org/abs/2405.01953 |