On some explicit integrals related to "fractal foothills"

TL;DR

This paper derives explicit integral formulas for partial loop counting functions in infinite random walks, focusing on returns to the origin, and explores their connections with Bernoulli polynomials.

Contribution

It introduces closed-form expressions for integrals related to partial loop counting functions, simplifying previous analyses and linking to Bernoulli polynomials.

Findings

Closed-form integral expressions for partial loop counting functions.

Connections established between loop counting functions and Bernoulli polynomials.

Applications to complete loop counting functions demonstrated.

Abstract

In the previous papers, we tried to analyze the complete loop counting functions that count all the loops in an infinite random walk represented by digits of a real number. In this paper, the consideration will be restricted to the partial loop counting functions $V$ that count the returns to the origin only. This simplification allows us to find closed-form expressions for various integrals related to $V$. Some applications to the complete loop counting functions, in particular, their connections with Bernoulli polynomials, are also provided.