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It is known that any rational abstract numeration system is faithfully, andeffectively, represented by an N-rational series. A simple proof of this resultis given which yields a representation of this series which in turn allows asimple computation of the value of words in this system and easy constructionsfor the recognition of recognisable sets of numbers. It is also shown thatconversely it is decidable whether an N-rational series corresponds to arational abstract numeration system.

On the enumerating series of an abstract numeration system