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Digital Sums and Functional Equations

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Published/Copyright: January 24, 2012
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Integers
From the journal Volume 12 Issue 1

Abstract.

Let S(n) denote the total number of digits `1' in the binary expansions of the integers between 1 and n-1. The Trollope–Delange formula is a classical result which provides an explicit representation for S(n) in terms of the continuous, nowhere differentiable Takagi function. Recently, connections have been established between digital sums such as S(n) and certain functional equations associated with the Takagi function and its relatives. In the present paper we explore such a connection to derive a new, simple proof for the Trollope–Delange formula as well as for some of its generalizations involving power and exponential sums.

Keywords.: Trollope–Delange Formula; Takagi Function; Digital Sums; Functional Equations
Received: 2010-08-30
Revised: 2011-06-14
Accepted: 2011-08-22
Published Online: 2012-01-24
Published in Print: 2012-February

© 2012 by Walter de Gruyter Berlin Boston

Articles in the same Issue

  1. Prelims
  2. Selmer's Multiplicative Algorithm
  3. Lucas Numbers and Determinants
  4. A Combinatorial Proof of Guo's Multi-Generalization of Munarini's Identity
  5. Enumeration of Triangles in Quartic Residue Graphs
  6. A remark on the Boros–Moll Sequence
  7. A Relation Between Triangular Numbers and Prime Numbers
  8. Some Remarks on a Paper of V. A. Liskovets
  9. A Combinatorial Proof of a Recursive Formula for Multipartitions
  10. Aliquot Cycles of Repdigits
  11. Digital Sums and Functional Equations
https://doi.org/10.1515/integ.2011.091
Keywords for this article
Trollope–Delange Formula; Takagi Function; Digital Sums; Functional Equations

Articles in the same Issue

  1. Prelims
  2. Selmer's Multiplicative Algorithm
  3. Lucas Numbers and Determinants
  4. A Combinatorial Proof of Guo's Multi-Generalization of Munarini's Identity
  5. Enumeration of Triangles in Quartic Residue Graphs
  6. A remark on the Boros–Moll Sequence
  7. A Relation Between Triangular Numbers and Prime Numbers
  8. Some Remarks on a Paper of V. A. Liskovets
  9. A Combinatorial Proof of a Recursive Formula for Multipartitions
  10. Aliquot Cycles of Repdigits
  11. Digital Sums and Functional Equations
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