2024/05/03 by Carton, Olivier, Yassawi, Reem
#11B85 #68Q45 #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.01953
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.