Mahler equations for Zeckendorf numeration
This work generalizes prior results in numeration systems, offering incremental theoretical insights for researchers in automata theory and formal languages.
The paper tackles the problem of relating sequences defined by Zeckendorf numeration to solutions of Z-Mahler equations, showing that Z-regular sequences correspond to solutions of isolating Z-Mahler equations and providing a counterexample for non-isolating cases.
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.