Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function

TL;DR

This paper derives an exact formula for the moments of the Minkowski question mark function using semi-regular continued fractions, providing a new approach that avoids explicit regular continued fractions.

Contribution

It introduces a novel formula for the moments of the Minkowski question mark function based on semi-regular continued fractions, independent of prior results.

Findings

Established a formula for the moments of the Minkowski question mark function

Connected the moments to semi-regular continued fractions

Provided a self-contained proof without relying on previous work

Abstract

This paper continues investigations on the integral transforms of the Minkowski question mark function. In this work we finally establish the long-sought formula for the moments, which does not explicitly involve regular continued fractions, though it has a hidden nice interpretation in terms of semi-regular continued fractions. The proof is self-contained and does not rely on previous results by the author.