Title:Mahler equations for Zeckendorf numeration
Speaker:Olivier Carton(University Paris Cité)
Abstract:
Let U = (u_n) be a Pisot numeration system. A sequence (f_n) taking values over a commutative ring R, possibly infinite, is said to be U-regular if there exists a weighted automaton which outputs f_n when it reads (n)_U. For base-q numeration, with q ∈ ℕ, q-regular sequences were introduced and studied by Allouche and Shallit, and they are a generalisation of q-automatic sequences (f_n), where f_n is the output of a deterministic automaton when it reads (n)_q. Becker, and also Dumas, made the connection between q-regular sequences, and q-Mahler type equations. In particular a q-regular sequence gives a solution to an equation of q-Mahler type, and conversely, the solution of an isolating, or Becker, equation of q-Mahler type is q-regular.
We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system~Z. 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. 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.
This is joint work with Reem Yassawi.
Time:10.22，15：00-16：00
Location: Zoom ID：993 8441 9798 Password：874560
