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_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.

39 pages, 1 figure (submitted). Note that the images of curves I_p are absent (they are contained in previous versions.) v4: Proofs in Section 3 completed

References in corpus (6)

• Fractal analysis for sets of non-differentiability of Minkowski's question mark function
• The moments of Minkowski question mark function: the dyadic period function
• Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function
• Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]
• Two-Dimensional Analogs of the Minkowski ?(x) Function
• Statistical properties of the Calkin--Wilf tree: real an p-adic distribution

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• Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
• Fourier-Stieltjes coefficients of the Minkowski question mark function
• Zeros and convergent subsequences of Stern polynomials
• The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I
• Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator
• Minkowski's question mark measure is UST--regular
• Errata and Addenda to Mathematical Constants