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arXiv · 1705.03857

Power-Sum Denominators

Bernd C. Kellner·Jonathan Sondow
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Abstract

The power sum $1^n + 2^n + \cdots + x^n$ has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in $x$ of degree $n+1$ with rational coefficients. Here we consider the denominators of these polynomials, and prove some of their properties. A remarkable one is that such a denominator equals $n+1$ times the squarefree product of certain primes $p$ obeying the condition that the sum of the base-$p$ digits of $n+1$ is at least $p$. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

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Bernd C. Kellner, Jonathan Sondow. 2017-05-10. Power-Sum Denominators. https://doi.org/10.4169/amer.math.monthly.124.8.695

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