Circular words and three applications: factors of the Fibonacci word, F-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,..

We introduce the notion of circular words with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word abaababaabaab . . .. Second, we present an expression of the sequence A004146 of [S] in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of p-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base p. Among such F-adic numbers, we give a characterization of the subset of those which are rational (that is: a root of an equation of the form qX = p, for integral values of p and q) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of qX = p, showing in particuler that it contains exactly q F -adic elements. Classically, a (finite) word is a finite sequence of elements (or letters) of a given set, the alphabet. Here, we mean by circular word a finite word w0 . . . wn in which the last letter, wn, is assumed to be followed by the first one, w0. This definition gives rise to interesting properties when circular words are assumed to be admissible, that is, made of letters in the alphabet {0, 1} without any two successive letters equal to 1. These properties derive from an underlying algebraic structure: the set of admissible circular words of fixed even length is an abelian group, which can be explicitely written as a product of finite monogenetic groups. One of the properties of circular admissible words of length 2l is that their cardinality cl is given by the sequence A004146 of [S] (which starts by 1, 5, 16, 45, 121, 320,. . . ), which has many important combinatorial properties (see [R]). The link between this sequence and admissible circular words appears also in the sequence of determinants of a sequence of linear operators we consider for our study. This fact gives rise to a formula that expresses each element of the sequence A004146 as an explicit product of expressions of the form 1− α− α, where α is a root of unity. 2010 Mathematics Subject Classification. Primary 68R15; Secondary 11A63, 11E95, 37B10.