## $$Pm$$ numbers, ambiguity, and regularity.(English)Zbl 0806.11007

The authors study the so-called pseudo-$$m$$-ray number system $$(Pm)$$, where integers are represented by sums $$\sum a_ i {{m^{i+1}-1} \over {m^ i-1}}$$, with $$a_ i=0,1,2, \dots,m$$ ($$m$$ is given $$>1$$). They show that the language consisting of the $$Pm$$ representation of the natural integers obtained by the greedy algorithm is regular. Furthermore they show that the language of the representations of the non-ambiguous integers is also regular (a “non-ambiguous” integer admits only one representation in this system).
Note that some of these representations are related to the sequence of parentheses occurring in the recursive definition of integers [see J.-P. Allouche, J. Bétréma and J. O. Shallit, RAIRO, Inf. Théor. Appl. 23, 235-249 (1988; Zbl 0691.68065)], but also (at least for $$q=2$$) to proofs of transcendence of values of the Carlitz zeta function [see J.-P. Allouche, Sémin. Théor. Nombres Bordx., II. Ser. 2, 103-117 (1990; Zbl 0709.11067), see also V. Berthé, Acta Arith. 66, 369-390 (1994) and Combinaisons linéaires de $$\zeta(s)/\pi^ s$$ sur $$\mathbb{F}_ q(x)$$, pour $$1\leq s\leq q-2$$, J. Number Theory (to appear)].
Note also that the $$Pm$$ representations are related to $$m$$-ary trees [H. A. Cameron, Extremal cost binary trees, Ph. D. Thesis, University of Waterloo (1991) and H. A. Cameron and D. Wood, Discrete Appl. Math. 55, 15-35 (1994)].
Finally “syms” (first line of the Abstract) should be replaced by “sums” and the usual French word for greedy algorithm is “algorithme glouton”.

### MSC:

 11A67 Other number representations 68Q45 Formal languages and automata

### Citations:

Zbl 0691.68065; Zbl 0709.11067
Full Text:

### References:

 [1] J.-P. ALLOUCHE, J. BETREMA and J. O. SHALLIT, Sur des points fixes de morphismes d’un monoïde libre, Informatique Théorique et Applications/Theoretical Informatics and Applications, 23, (3), 1989, pp. 235-249. Zbl0691.68065 MR1020473 · Zbl 0691.68065 [2] H. A. CAMERON, Extremal Cost Binary Trees, Ph. D. thesis, University of Waterloo, 1991. [3] H. A. CAMERON and D. WOOD, The maximal path length of binary trees, Discrete Applied Mathematics, to appear, 1993. Zbl0821.68094 · Zbl 0821.68094 [4] A. S. FRAENKEL, Systems of numeration, The American Mathematical Monthly, 92, (2), 1985, pp. 105-114. Zbl0568.10005 MR777556 · Zbl 0568.10005 [5] J. SHALLIT, Numeration Systems, linear recurrences, and regular sets, Technical Report CS-91-32, University of Waterloo, July 1991. · Zbl 0810.11006
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