Integrality and some evaluations of odd multiple harmonic sums

TL;DR

This paper investigates the integrality of odd multiple harmonic sums and their star variants, proving most are non-integers except for trivial cases, and provides explicit evaluations for depth-one sums.

Contribution

It extends previous results by analyzing odd multiple harmonic sums and star sums, establishing their non-integrality except in trivial cases, and offers explicit evaluations for depth-one sums.

Findings

Most odd multiple harmonic sums are not integers.

Explicit evaluations are provided for depth-one sums.

Non-integrality holds except for trivial cases.

Abstract

In 2015, S. Hong and C. Wang proved that none of the elementary symmetric functions of $1,1/3,\ldots,1/(2n-1)$ is an integer when $n\geq 2$. In 2017, Kh. Pilehrood, T. Pilehrood and R. Tauraso proved that the multiple harmonic sums $H_n(s_1,\ldots,s_r)$ are never integers with exceptions of $H_1(s_1)=1$ and $H_3(1,1)=1$. They also proved that the multiple harmonic star sums are never integers when $n\geq 2$. In this paper, we consider the odd multiple harmonic sums and the odd multiple harmonic star sums and show that none of these sums is an integer with exception of the trivial case. Besides, we give evaluations of the odd (alternating) multiple harmonic sums with depth one.