Algebraic Aspects of B-regular Series

This paper concerns power series of an arithmetic nature that arise in the analysis of divide-and-conquer algorithms. Two key notions are studied: that of B-regular sequence and that of Mahlerian sequence with their associated power series. Firstly we emphasize the link between rational series over the alphabet fx0;x1;:::;xB?1g and B-regular series. Secondly we extend the theorem of Christol, Kamae, Mend es France and Rauzy about automatic sequences and algebraic series to B-regular sequences and Mahlerian series. We develop here a constructive theory of B-regular and Mahlerian series. The examples show the ubiquitous character of B-regular series in the study of arithmetic functions related to number representation systems and divide-and-conquer algorithms. The interest of 2-regular sequences comes from their presence in many problems which touch upon the binary representation of integers or divide-and-conquer algorithms, like sum-of-digits function, number of odd binomial coef-cients, Josephus problem, mergesort, Euclidean matching or comparison networks. This explains why we study B-regular sequences that formalize the sequences which are solutions of certain diierence equations of the divide-and-conquer type. In other words we want to show that B-regular series (i.e. generating functions of B-regular sequences) are as important in computer science as rational functions are common in mathematics. Many properties of B-regular sequences like closure properties or growth properties have been etablished by Allouche and Shallit. In particular they showed that there is a link between B-regular sequences and rational series in the sense of formal language theory. The transition from one to another uses the B-ary representation of integers. There is already a long tradition about recognizable sets and automatic sequences. The link provides us with the well known machinery of rational series and the rst part of the paper is devoted to the illustration of its use. For example we introduce the Hankel matrix of a regular series. This is the practical way to nd the rank of a regular series, to exhibit minimal recurrence relations or to build up linear representations. In the second part we compare B-regular series and Mahlerian series. Our goal is to extend the theorem of Christol, Kamae, Mend es France and Rauzy 6], which asserts that q-automatic series with coeecients in the nite eld F q are exactly algebraic series. To that purpose we introduce a more general notion of Mahlerian series. We prove in particular that B-regular series are Mahlerian series.