A $q$-weighted analogue of the Trollope-Delange formula

Let $s(n)$ denote the number of "$1$"s in the dyadic representation of a positive integer $n$ and sequence $S(n)=s(1)+s(2)+\cdots +s(n-1)$. The Trollope-Delange formula is a classic result that represents the sequence $S$ in terms of the Takagi function. This work extends the result by introducing a $q$-weighted analog of $s(n)$, deriving a variant of the Trollope-Delange formula for this generalization. We show that for $1\mathrm{/}2<\mathrm{\mid }q\mathrm{\mid }<1$, nondifferentiable Takagi-Landsberg functions appear, whereas for $\mathrm{\mid }q\mathrm{\mid }>1$, the resulting functions are differentiable almost everywhere. We further show how the result can be used to find limiting curves describing fluctuations in the ergodic theorem for the dyadic odometer.