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

TL;DR

This paper introduces a family of functions related to the Minkowski question mark function, providing explicit series representations for the dyadic period function and its moments, revealing new structural insights.

Contribution

It defines a family of distributions F_p(x) extending the question mark function and proves the dyadic period function can be expressed as an infinite series of rational functions.

Findings

Dyadic period function expressed as an infinite series of rational functions

Moments of F_p(x) satisfy a three-term functional equation

Introduces a new family of distributions related to F(x)

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_p(x) for Re p>=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_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.