Euler sums and integrals of polylogarithm functions

Xu, Ce; Yan, Yuhuan; Shi, Zhijuan

doi:10.1016/j.jnt.2016.01.025

Cited by 62 publications

(63 citation statements)

References 9 publications

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“…were introduced by Kölbig [28,29] in 1980s, and reconsidered recently by us [42,43,46,53] to study the Euler sums. From these works, we know that…”

Section: Some Special Cases Of Theorems 21 and 22mentioning

confidence: 99%

“…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%

“…S r 2 ,2r are reducible to linear sums and polynomials in zeta values, so the remark in the corollary also holds. Note that (3.18) has been presented in[53, Eq. (3.40)].Example 3.1.…”

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confidence: 99%

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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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“…were introduced by Kölbig [28,29] in 1980s, and reconsidered recently by us [42,43,46,53] to study the Euler sums. From these works, we know that…”

Section: Some Special Cases Of Theorems 21 and 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Explicit formulas of Euler sums via multiple zeta values

Wang

2020

Journal of Symbolic Computation

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“…However, there are also many nonlinear Euler sums which need not only zeta values but also linear sums. Namely, many nonlinear Euler sums are reducible to polynomials in zeta values and to linear sums (see [3,5,15,18,27,30]). For instance, in 1995, Borwein et al [5] showed that the quadratic sums S 1 2 ,q can reduce to linear sums S 2,q and polynomials in zeta values.…”

Section: Introductionmentioning

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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“…The relationship between the values of the Riemann zeta function and the classical Euler sums W 0 (m; p) (or S m;p ) has been studied by many authors (for example, see [7][8][9][10][11][12][13][14][15][16] and the references therein). So far, surprisingly little work has been done on q-analogues of Euler sums.…”

Section: Introductionmentioning

confidence: 99%

Some results on q-harmonic number sums

2018

Adv Differ Equ

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In this paper, we establish some relations involving q-Euler type sums, q-harmonic numbers and q-polylogarithms. Then, using the relations obtained with the help of q-analog of partial fraction decomposition formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit formulas for several classes of q-harmonic sums in terms of q-polylogarithms and q-harmonic numbers. The given representations are new.MSC: 05A30; 65B10; 33D05; 11M99; 11M06; 11M32

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Euler sums and integrals of polylogarithm functions

Cited by 62 publications

References 9 publications

Explicit formulas of Euler sums via multiple zeta values

Explicit formulas of Euler sums via multiple zeta values

Some evaluation of cubic Euler sums

Some results on q-harmonic number sums

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