Extension of Summation Formulas involving Stirling series

Schumacher, Raphael

doi:10.48550/arxiv.1605.09204

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Convergent Version Of the Euler-maclaurin Summation Formula1

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2021

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“…We believe that they will also have applications in physics [18] such as the extended version of Faulhaber's formula [19,20]. With the universal technique, explained in this paper, one can obtain other summation formulas of this type [21,22], as for example with Theorem 10 and equation (28) we obtain: (68)…”

Section: Discussionmentioning

confidence: 97%

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The Generalization of Faulhaber's Formula to Sums of Arbitrary Complex Powers

Schumacher

2021

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In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers m ∈ C. These summation formulas for sums of the form ⌊x⌋ k=1 k m and n k=1 k m , where x ∈ R + and n ∈ N, are based on a series acceleration involving Stirling numbers of the first kind. While it is well-known that the corresponding expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent.

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Section: Discussionmentioning

confidence: 97%

“…A generalization of Faulhaber's formula for alternating sums can be found in [22] and an extended form of it is given for x ∈ R + by:…”

Section: Convergent Version Of the Euler-maclaurin Summation Formulamentioning

confidence: 99%