Abstract
The possible polynomial expressions for sums of powers of integers multiplied by an exponential term are investigated. We explicitly give factorization of these polynomials in terms of the roots of Apostol-Bernoulli polynomials. As a special case, alternating sums of powers of integers are also considered, and some new polynomial expressions are given.
Highlights
The sums of the form n∑−1 ak−1δa a=0 for δ = ±1 and k ≥ 2 an integer have been studied over the centuries
The classical Faulhaber theorem states that for an even integer k ≥ 2 the sum n∑−1 ak−1 a=0 is a polynomial in n(n−1)/2
One may consider sums of fixed powers of the terms {a + ib}ni=−01, which were studied in [4, 5, 7]. These types of sums are closely related to Bernoulli and Euler polynomials/numbers
Summary
A generalization of this result, namely an explicit form for the sum n∑−1 (−1)n−a(y + a)2m, a=0 is given in terms of both Euler and Apostol–Bernoulli polynomials (see Theorem 2.3 in [7] and Equation (1.11) in [13]). Another direction is the study of combinatorial properties of q -analogues for sums of powers, namely the sums obtained by replacing a by [a] := (1 − qa)/(1 − q) , where q can be seen as indeterminate. Recall that by Theorem 1 βk(x, −1) ∈ Z[1/2][x]
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