Manfred Krüppel

Functional Equations for Knopp Functions and Digital Sums

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 65, 85 - 101 (2010)

MSC: 11A63   Radix representation; digital problems For metric results, see 11K16
  39B22   Equations for real functions [See also 26A51, 26B25]
  26A27   Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
  42A16   Fourier coefficients, Fourier series of functions with special properties, special Fourier series For automorphic theory, see mainly 11F30

Abstract:   The well-known Delange formula expressed the usual sum-of-digits function to a basis $q\ge2$ by means of a continuous, nowhere differentiable function. The aim of this paper is to clarify the actually reason for this phenomenon. For this we show that specific Knopp functions satisfy functional equations which allow to calculate, for any positive integer $n$, the number of times of digits in the $q$-ary representation of $n$ which are equal to a fixed $m\in\{1,\ldots,q-1\}$. By linear combination for arbitrary Knopp functions we get functional equations contained certain digital sums. These functional equations imply sum formulas for certain digital sums. Simple examples are the formula of Delange for the usual sum-of-digits function and a formula for the number of zeros.

Keywords:   Knopp functions, functional equations, digital sums, Fourier expansion
Seite generiert am 15.12.2010,   19:26   Uhr