  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.
