| 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.