Power-sum denominators. (English) Zbl 1391.11052

Summary: The power sum \(1^n+2^n+\dots+x^n\) 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.


11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials


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