The generalization of Faulhaber's formula to sums of non-integral powers

McGown, Kevin J.; Parks, Harold R.

doi:10.1016/j.jmaa.2006.08.019

Cited by 16 publications

(10 citation statements)

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“…This linear recurrence is direct result of second order binomial transform of M 1 (n, k) over n. (7) Linear recurrence, for each n > k…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Series Representation of Power Function

Kolosov¹

2018

Preprint

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In this paper described numerical expansion of natural-valued power function x n , in point x = x 0 , where n, x 0 -positive integers. Applying numerical methods, the calculus of finite differences, particular pattern, that is sequence A287326 in OEIS, which shows the expansion of perfect cube n as row sum over k, 0 ≤ k ≤ n − 1 is generalized, obtained results are applied to show expansion of monomial n 2m+1 , m = 0, 1, 2, ..., N. Additionally, relation between Faulhaber's sum n m and finite differences of power are shown in section 4.

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“…This linear recurrence is direct result of second order binomial transform of M 1 (n, k) over n. (7) Linear recurrence, for each n > k…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Series Representation of Power Function

Kolosov¹

2018

Preprint

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“…Definition 4. (Bernoulli polynomials and Bernoulli numbers) [1], [12], [13], [14] We define for n ∈ N 0 the n-th Bernoulli polynomial B n (x) via the following exponential generating function [1] as…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

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

Schumacher

2021

Preprint

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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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“…Sum of integer exponent of natural numbers is algebraically not very intricate, and sum for specific integer exponents (especially 1,2,3) is easily available on introductory algebra books. Calculating higher exponents as well as generalizing the formula [4,12] have been already done. This paper aims to present an algebraic approach to this problem to the readers with minimal knowledge of number theory, combinatorics, calculus and linear algebra.…”

Section: Introductionmentioning

confidence: 99%

Linear Algebraic Approach to Formulate A New Recurrence Relation for Bernoulli Numbers from the Power-Sum of Natural Numbers with Experiments on Pedagogy

2021

Punjab Univ. j. math.

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In this article, a new recurrence relation formula for Bernoulli numbers have been derived, and sum of integer exponents of natural numbers has been revisited from this novel perspective. Some interesting pedagogical experiments on wording and presentation of mathematical derivation have been attempted, and development from first principle have been undertaken in line with this experimental approach.

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The generalization of Faulhaber's formula to sums of non-integral powers

Abstract: A formula for the sum of any positive-integral power of the first N positive integers was published by Johann Faulhaber in the 1600s. In this paper, we generalize Faulhaber's formula to non-integral complex powers with real part greater than −1.

Cited by 16 publications

References 1 publication

Series Representation of Power Function

Series Representation of Power Function

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

Linear Algebraic Approach to Formulate A New Recurrence Relation for Bernoulli Numbers from the Power-Sum of Natural Numbers with Experiments on Pedagogy

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