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

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.