MINKOWSKI QUESTION MARK FUNCTION AND ITS GENERALIZATIONS, ASSOCIATED WITH p−CONTINUED FRACTIONS: FRACTALS, EXPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION AND MOMENTS

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.