The Minkowski question mark function: explicit series for the dyadic period function and moments

Abstract:

Previously, several natural integral transforms of the Minkowski question mark function $F(x)$ were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about $F(x)$. One of them, the dyadic period function $G(z)$, was defined as a Stieltjes transform. In this paper we introduce a family of “distributions” $F_{\textsf {p}}(x)$ for $\Re \sf p\geq 1$, such that $F_{1}(x)$ is the question mark function and $F_{2}(x)$ is a discrete distribution with support on $x=1$. We prove that the generating function of moments of $F_\textsf {p}(x)$ satisfies the three-term functional equation. This has an independent interest, though our main concern is the information it provides about $F(x)$. This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.