  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.
