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Mathematics of Computation

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

Author: Giedrius Alkauskas
Journal: Math. Comp. 79 (2010), 383-418
MSC (2000): Primary 11A55, 26A30, 32A05; Secondary 40A15, 37F50, 11F37
Published electronically: May 12, 2009
Corrigendum: Math. Comp. 80 (2011), 2445-2454
MathSciNet review: 2552232
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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_{{\sf 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_{\sf 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.

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Additional Information

Giedrius Alkauskas
Affiliation: The Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, Lithuania and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Keywords: The Minkowski question mark function, the dyadic period function, three-term functional equation, analytic theory of continued fractions, Julia sets, the Farey tree
Received by editor(s): September 15, 2008
Received by editor(s) in revised form: January 17, 2009
Published electronically: May 12, 2009
Article copyright: © Copyright 2009 American Mathematical Society