[Paper Review] Minkowski question mark function and its generalizations, associated with p-continued fractions: fractals, explicit series for the dyadic period function and moments

Giedrius Alkauskas|arXiv (Cornell University)|May 12, 2008

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper introduces a one-parameter family of distributions Fp(x) for ℜp ≥ 1, generalizing the Minkowski question mark function F(x) (where F1(x) = F(x)). It derives explicit series for the dyadic period function G(z) and shows that the generating function of moments of Gp(z) satisfies a three-term functional equation, enabling closed-form expressions for the moments of F(x).

ABSTRACT

Previously, several natural integral transforms of 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 via certain transcendental integral. In this paper we introduce a family of “distributions ” Fp(x) for ℜ p ≥ 1, such that F1(x) is the question mark function and F2(x) is a discrete distribution with support on x = 1. Further, all the aforementioned integral transforms are calculated for such p, and the generating function of moments of G p(z) 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 certain explicit series for G(z). This also solves the problem in expressing the moments of F(x) in closed form.

Motivation & Objective

• To generalize the Minkowski question mark function F(x) into a one-parameter family Fp(x) for ℜp ≥ 1, with F1(x) recovering the original function.
• To extend previously introduced integral transforms—particularly the dyadic period function G(z)—to the generalized Fp(x) framework.
• To derive explicit series representations for G(z) using the generalized framework.
• To establish that the generating function of moments of Gp(z) satisfies a three-term functional equation, enabling closed-form moment expressions.
• To solve the longstanding problem of expressing the moments of F(x) in closed form using this new approach.

Proposed method

• Introduces a family of distributions Fp(x) for ℜp ≥ 1, with F1(x) being the standard Minkowski question mark function and F2(x) a discrete measure at x = 1.
• Defines generalized integral transforms, including the dyadic period function Gp(z), via transcendental integrals analogous to the original G(z).
• Derives the generating function of moments for Gp(z) and shows it satisfies a three-term functional equation, a key structural property.
• Uses the functional equation of the moment generating function to derive explicit series representations for G(z) in the original case (p = 1).
• Applies the generalized framework to recover and express the moments of F(x) in closed form through the moment-generating structure of Gp(z).
• Establishes uniqueness and regularity conditions for the integral transforms, ensuring intrinsic encoding of F(x)'s properties.

Experimental results

Research questions

• RQ1 How can the Minkowski question mark function F(x) be generalized to a one-parameter family Fp(x) for ℜp ≥ 1, preserving key functional properties?
• RQ2 What is the explicit form of the dyadic period function G(z) in the generalized setting, and can it be expressed via convergent series?
• RQ3 Does the generating function of moments for Gp(z) satisfy a three-term functional equation, and what does this imply for moment computation?
• RQ4 Can the moments of the original F(x) be expressed in closed form using the generalized framework?
• RQ5 What intrinsic structural properties of F(x) are encoded in the generalized integral transforms and their functional equations?

Key findings

• The paper successfully generalizes the Minkowski question mark function to a one-parameter family Fp(x) for ℜp ≥ 1, with F1(x) recovering the original function.
• Explicit series representations for the dyadic period function G(z) are derived using the generalized framework, providing new analytical tools.
• The generating function of moments for Gp(z) satisfies a three-term functional equation, which is a central structural result.
• This functional equation enables the derivation of closed-form expressions for the moments of F(x), solving a long-standing open problem.
• The generalized integral transforms are uniquely characterized by regularity conditions and functional equations, ensuring they encode intrinsic information about F(x).
• The approach establishes a direct link between the functional equation of the moment generating function and the explicit computation of moments, offering a novel computational pathway.