Positional number systems with digits forming an arithmetic progression

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• 作者：Clemens Heuberger (1)
  Helmut Prodinger (2)
  Stephan G. Wagner (2)
  
• 关键词：Number system ; Digit formul? ; Digit frequencies ; Mellin ; Perron summation formula ; 11A63
• 刊名：Monatshefte f眉r Mathematik
• 出版年：2008
• 出版时间：December 2008
• 年：2008
• 卷：155
• 期：3-4
• 页码：349-375
• 全文大小：412KB
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• 作者单位：Clemens Heuberger (1)
  Helmut Prodinger (2)
  Stephan G. Wagner (2)
  
  1. Institut für Mathematik B, Technische Universit?t Graz, Graz, Austria
  2. Department of Mathematical Sciences, University of Stellenbosch, Matieland, South Africa
  
• ISSN：1436-5081

文摘

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 formul? 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.