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Kellner, Bernd C.; Sondow, Jonathan

Power-sum denominators. (English) Zbl 1391.11052

Am. Math. Mon. 124, No. 8, 695-709 (2017).
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.

Cited in 9 Documents

MSC:

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

Keywords:

Bernoulli polynomials

Software:

OEIS
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\textit{B. C. Kellner} and \textit{J. Sondow}, Am. Math. Mon. 124, No. 8, 695--709 (2017; Zbl 1391.11052)
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Online Encyclopedia of Integer Sequences:

a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).
a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.
a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n), k = 0..n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

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