Evaluations of Euler-Type Sums of Weight $$\le 5$$≤5

Xu, Ce

doi:10.1007/s40840-018-00715-3

Cited by 9 publications

(3 citation statements)

References 28 publications

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“…where the special values for the dilogarithm and trilogarithm given in ( 27) have been used. Alternative derivations for ( 12) and ( 13) can be found in [11,43].…”

Section: Some Series As Consequences Of Lemmas 21 and 22 And Theorem 23mentioning

confidence: 99%

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Stewart¹

2020

Tatra Mountains Mathematical Publications

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In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung \sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.

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“…where the special values for the dilogarithm and trilogarithm given in ( 27) have been used. Alternative derivations for ( 12) and ( 13) can be found in [11,43].…”

Section: Some Series As Consequences Of Lemmas 21 and 22 And Theorem 23mentioning

confidence: 99%

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Stewart¹

2020

Tatra Mountains Mathematical Publications

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“…An excellent overview about these works is found in the paper of Flajolet and Salvy 1 and in citations therein. Further work on similar types of Euler sums can e found, for example, in [5][6][7] . Although, only few sums of the type…”

Section: Introductionmentioning

confidence: 99%

On an infinite number of new families of odd-type Euler sums

Braun¹,

Romberger²,

Bentz³

2021

Preprint

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“…It must be mentioned that such integrals are usually attacked in literature via Euler sums like in [3], [1] and [6]: each integral are converted in a combination of Euler sums with known values. However, virtually all discussions available are with weight ď 5.…”

Section: Introductionmentioning

confidence: 99%

Linear relations between logarithmic integrals of high weight and some closed-form evaluations

2019

Preprint

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The focus of our investigation will be integrals of form ş 1 0 log a p1´xq log b x log c p1`xq{f pxqdx, where f can be either x, 1 ´x or 1 `x. We show that these integrals possess a plethora of linear relations, and give systematic methods of finding them. In lower weight cases, these linear relations yields remarkable closed-forms of individual integrals; in higher weight, we discuss the Q-dimension that such integrals span. Our approach will not feature Euler sums.

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Evaluations of Euler-Type Sums of Weight $$\le 5$$≤5

Abstract: Let p, p 1 , . . . , p m be positive integers with p 1

Cited by 9 publications

References 28 publications

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

On an infinite number of new families of odd-type Euler sums

Linear relations between logarithmic integrals of high weight and some closed-form evaluations

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