Positional number systems with digits forming an arithmetic progression

Abstract

A novel digit system that arises in a natural way in a graph-theoretical problem is studied. It is defined by a set of positive digits forming an arithmetic progression and, necessarily, a complete residue system modulo the base b. Since this is not enough to guarantee existence of a digital representation, the most significant digit is allowed to come from an extended set. We provide explicit formulæ for the j th digit in such a representation as well as for the length. Furthermore, we study digit frequencies and average lengths, thus generalising classical results for the base-b representation. For this purpose, an appropriately adapted form of the Mellin-Perron approach is employed. © 2008 Springer-Verlag.