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The denominators of power sums of arithmetic progressions. (English) Zbl 1423.11029

The authors study the denominators of polynomials that represent the power sums of arithmetic progressions: \[ {\mathcal S}_{m,r}^n(x)=\sum_{k=0}^{x-1}(km+r)^n=r^n+(m+r)^n+\dots+((x-1)m+r)^n. \] They extend their earlier results on the case of power sum’s (when \(r=0,m=1\)). Specially, they give a simple explicit criterion for the integrality of the coefficients of these polynomials, and show further applications about the sequence of denominators of the Bernoulli polynomials.

MSC:

11B25 Arithmetic progressions
11B68 Bernoulli and Euler numbers and polynomials

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Online Encyclopedia of Integer Sequences:

a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.
Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.
a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).
a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).
a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).
a(n) = b(2*n)/b(2*n+1) where b(n) = denominator(Bernoulli_{n}(x)).
Indices k such that A195441(k) = A195441(k+1).
Numbers that appear in A195441 at least once for two consecutive indices.
a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.
Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).
Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.
Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.
Positive integers k such that the fifth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

References:

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