On harmonic numbers and nonlinear Euler sums

Xu, Ce; Cai, Yulin

doi:10.1016/j.jmaa.2018.06.036

Cited by 13 publications

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“…In [18,21,23,24], we obtain numerous results of some alternating Euler sums of weight ≤ 6. We can use these results to find some nice evaluations of integral of logarithms.…”

Section: Some Results On Integral Of Logarithmsmentioning

confidence: 99%

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

2019

Bull. Malays. Math. Sci. Soc.

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“…In [18,21,23,24], we obtain numerous results of some alternating Euler sums of weight ≤ 6. We can use these results to find some nice evaluations of integral of logarithms.…”

Section: Some Results On Integral Of Logarithmsmentioning

confidence: 99%

“…Hence, in this section, we will give many closed form representations of logarithms' integrals. By using Lemma 2.4, 2.5 and formula (3.6) with the help of results of references [18,21,23,24], the following identities are easily derived…”

Section: Some Results On Integral Of Logarithmsmentioning

confidence: 99%

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

2019

Bull. Malays. Math. Sci. Soc.

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“…Next, we give a simple case. From [3,26,31], we know that S1 2,3 = ∞ n=1H n H Hence, the formula (5.12) can be rewritten as S13…”

Section: Some Examples and Corollariesmentioning

confidence: 99%

Some evaluation of cubic Euler sums

2018

Journal of Mathematical Analysis and Applications

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P. Flajolet and B. Salvy [15] prove the famous theorem that a nonlinear Euler sum S i 1 i 2 ···ir,q reduces to a combination of sums of lower orders whenever the weight i 1 + i 2 + · · · + i r + q and the order r are of the same parity. In this article, we develop an approach to evaluate the cubic sums S 1 2 m,p and S 1l 1 l 2 ,l 3 . By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums S 1 2 m,m and S 1(2l+1) 2 ,2l+1 are reducible to zeta values, quadratic and linear sums. Moreover, we prove that the two combined sums involving multiple zeta values of depth fourcan be expressed in terms of multiple zeta values of depth ≤ 3, here 2 ≤ m 1 , m 2 , m 3 ∈ N. Finally, we evaluate the alternating cubic Euler sums S13 ,2r+1 and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.

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“…Recently, rapid progress has been made in this field. Using the Bell polynomials, generating functions, integrals of special functions, multiple zeta (star) values, the Stirling sums and the Tornheim type series, we study the (alternating) Euler sums systematically [42][43][44][46][47][48][49][50][51]53,54]. As a consequence, the evaluation of all the unknown Euler sums up to the weight 11 are presented, and a basis of Euler sums of weight 3 ≤ w ≤ 11 is…”

Section: Introductionmentioning

confidence: 99%

“…Using the tables of alternating MZVs given in [5], we have computed all the alternating Euler sums of weight w ≤ 6. The evaluations of some alternating Euler sums of weight w ≤ 5 can be found in [44,[47][48][49]54], so we list in the following…”

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Explicit formulas of Euler sums via multiple zeta values

Wang

2020

Journal of Symbolic Computation

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Flajolet and Salvy pointed out that every Euler sum is a Q-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein-Borwein-Girgensohn theorems, and the Flajolet-Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of S r m ,q , Srm ,q , S r m ,q and Srm ,q , are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight w ≤ 11 for non-alternating case and of weight w ≤ 6 for alternating case.

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On harmonic numbers and nonlinear Euler sums

Cited by 13 publications

References 50 publications

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

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

Some evaluation of cubic Euler sums

Explicit formulas of Euler sums via multiple zeta values

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