Mahler equations for Zeckendorf numeration

A Pisot numeration system [Formula: see text] for [Formula: see text] is one where the sequence of positive integers [Formula: see text], with [Formula: see text], is generated by a recurrence whose polynomial is the minimal polynomial of a Pisot number. The Zeckendorf numeration Z is the simplest such example. We define generalized equations of Z-Mahler type, and 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 generalization 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. Our results can be generalized to numeration systems generated by recurrences whose characteristic polynomial is the minimal polynomial of a Pisot number.