Iterated harmonic numbers
Abstract
The harmonic numbers are the sequence 1, 1+1/2, 1+1/2+1/3, ... Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the harmonic numbers. The jth iterated harmonic numbers are a sequence of rational numbers that nests the previous sequences and relates in a similar way to the sequence of the jth iterate of the natural logarithm of positive integers. The analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including a generalization of Euler's constant. We reproduce the proof that only the first harmonic number is an integer and, providing some numeric evidence for the cases j = 2 and j = 3, conjecture that the same result holds for all iterated harmonic numbers. We also review another proposed generalization of harmonic numbers.
Cite
@article{arxiv.2303.02832,
title = {Iterated harmonic numbers},
author = {J Marshall Ash and Michael Ash and Rafael Ash and Ángel Plaza},
journal= {arXiv preprint arXiv:2303.02832},
year = {2023}
}
Comments
13 pages, 2 figures