On some explicit integrals related to "fractal foothills"
Combinatorics
2025-08-26 v2 Classical Analysis and ODEs
Functional Analysis
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 that count the returns to the origin only. This simplification allows us to find closed-form expressions for various integrals related to . Some applications to the complete loop counting functions, in particular, their connections with Bernoulli polynomials, are also provided.
Cite
@article{arxiv.2109.02741,
title = {On some explicit integrals related to "fractal foothills"},
author = {Anton A Kutsenko},
journal= {arXiv preprint arXiv:2109.02741},
year = {2025}
}
Comments
Formula (6) can simplify (3) and (4). The proof of the results is not unique, I chose exotic ones