Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers

TL;DR

This paper proves that multiple reciprocal sums and star sums of polynomials with nonnegative integer coefficients are almost never integers, extending previous results and identifying specific cases where they are integers.

Contribution

The paper generalizes non-integrality results for reciprocal sums of polynomials, covering higher degrees and multiple sums, and characterizes the unique cases where these sums are integers.

Findings

Most reciprocal sums are not integers for polynomials with degree at least 2.

The sums are integers only when f(x) = x^m with n=k=1.

Special case f(x)=2x-1 yields non-integer sums except at n=1.

Abstract

Let $n$ and $k$ be integers such that $1\le k\le n$ and $f(x)$ be a nonzero polynomial of integer coefficients such that $f(m)\ne 0$ for any positive integer $m$. For any $k$-tuple $\vec{s}=(s_1, ..., s_k)$ of positive integers, we define $$H_{k,f}(\vec{s}, n):=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\le n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}}$$ and $$H_{k,f}^*(\vec{s}, n):=\sum\limits_{1\leq i_{1}\leq \cdots\leq i_{k}\leq n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}}.$$ If all $s_j$ are 1, then let $H_{k,f}(\vec{s}, n):=H_{k,f}(n)$ and $H_{k,f}^*(\vec{s}, n):=H_{k,f}^*(n)$. Hong and Wang refined the results of Erd\"{o}s and Niven, and of Chen and Tang by showing that $H_{k,f}(n)$ is not an integer if $n\geq 4$ and $f(x)=ax+b$ with $a$ and $b$ being positive integers. Meanwhile, Luo, Hong, Qian and Wang established the similar result when $f(x)$ is of nonnegative integer coefficients and of degree no less than two. For any $k$-tuple $\vec{s}=(s_1, ..., s_k)$ of positive integers, Pilehrood, Pilehrood and Tauraso proved that $H_{k,f}(\vec{s},n)$ and $H_{k,f}^*(\vec{s},n)$ are nearly never integers if $f(x)=x$. In this paper, we show that if $f(x)$ is a nonzero polynomial of nonnegative integer coefficients such that either $\deg f(x)\ge 2$ or $f(x)$ is linear and $s_j\ge 2$ for all integers $j$ with $1\le j\le k$, then $H_{k,f}(\vec{s}, n)$ and $H_{k,f}^*(\vec{s}, n)$ are not integers except for the case $f(x)=x^{m}$ with $m\geq1$ being an integer and $n=k=1$, in which case, both of $H_{k,f}(\vec{s}, n)$ and $H_{k,f}^*(\vec{s}, n)$ are integers. Furthermore, we prove that if $f(x)=2x-1$, then both $H_{k,f}(\vec{s}, n)$ and $H_{k,f}^*(\vec{s}, n)$ are not integers except when $n=1$, in which case $H_{k,f}(\vec{s}, n)$ and $H_{k,f}^*(\vec{s}, n)$ are integers. The method of the proofs is analytic and $p$-adic.