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This paper extends some results of Allouche and Shallit for q-regular sequences to numeration systems in algebraic number fields and to linear numeration systems. We also construct automata that perform addition and multiplication by a fixed number. 1 Introduction A sequence is called q-automatic if its n-th term can be generated by a finite state machine from the q-ary digits of n. The concept of automatic sequences was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic property of automatic sequences: a sequence with values in a finite field of characteristic p is p-automatic if and only if the corresponding power series is algebraic over the field of rational functions over this finite field. A brief survey on this subject is given in [2], see also [10]. Some generalizations of this concept were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence has to t..

Year: 2000

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