On the Sum of Consecutive Kth Powers

Nunemacher, Jeffrey; Young, Robert M.

doi:10.2307/2689348

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Proof Of the Recursive Formula1

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“…Moreover, the remaining coefficients a k−1,r are connected to the Bernoulli numbers through the relation (see, for example, [13])…”

Section: Proof Of the Recursive Formulamentioning

confidence: 99%

A determinant formula for sums of powers of integers

Cereceda¹

2014

imf

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In this note, we first derive a recursive formula for the sum of powers S k (n) = 1 k + 2 k + · · · + n k , with k and n non-negative integers. We then apply it to establish, via Cramer's rule, an explicit determinant formula for S k (n) involving the Bernoulli numbers and the binomial (n + 1 2 ). Evaluating the determinant gives us directly S k (n) in the form of the so-called Faulhaber polynomial, namely as a sum of even or odd powers of (n + 1 2 ). Furthermore, a connection with Hessenberg matrices is shown.

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“…Moreover, the remaining coefficients a k−1,r are connected to the Bernoulli numbers through the relation (see, for example, [13])…”

Section: Proof Of the Recursive Formulamentioning

confidence: 99%