On some explicit integrals related to "fractal mountains"

TL;DR

This paper explores the fractal-like structure of loop counting functions related to digital walks, providing explicit integral formulas, Fourier series, and connections to special functions, with implications for analyzing self-avoiding walks.

Contribution

It introduces explicit integral representations and closed-form expressions for loop counting functions, linking them to rational functions, special functions, and continued fractions.

Findings

Integrals of the form ∫ x^A U(x)^B dx can be expressed using rational functions.

The integral ∫ x^A U(x) dx admits closed-form solutions.

Fourier series for U(x) are explicitly computed.

Abstract

Loop counting functions $U(x)$ estimate the number of "weighted" loops in a digital representation of $x\in[-1,1]$. Roughly speaking, each $x$ is considered as an infinite walk, where the steps of the walk correspond to digits of $x$. The graph of loop counting functions $U$ has a fractal structure that resembles complex mountain landscapes. In some sense, $U$ allows us to look at random walks globally. These functions may be helpful in the analysis of some hard problems related to the distribution of self-avoiding random walks (SAW) in a multi-dimensional case since SAW closely relate to zeros of $U(x)$. We note here that $U(x)$ can be naturally extended to a multidimensional argument $x$. In this article, the focus will be on some analytic aspects. It will be shown that integrals $\int x^AU(x)^Bdx$ with non-negative integers $A$ and $B$ can be expressed in terms of integrals of rational functions with integer coefficients. Moreover, it will be shown that $\int x^A U(x)dx$ admits closed-form expressions. Fourier series for $U$ is also computed. Finally, we discuss some connections with special functions and generalized continued fractions, and other perspectives.