Evaluations of nonlinear Euler sums of weight ten

Xu, Ce

doi:10.48550/arxiv.1703.00254

Cited by 4 publications

(3 citation statements)

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“…The nonlinear Euler sums, i.e., S π,q with π having two or more parts, are more complicated. Such sums were already considered in [3,23,43,46,47,49,51,52]. In [23], Flajolet and Salvy gave an algorithm for reducing S π 1 π 2 ,q to linear Euler sums when π 1 + π 2 + q is even and π 1 , π 2 , q > 1 (see Theorem 4.2 in the reference [23].…”

Section: Harmonic Numbers and Euler Sumsmentioning

confidence: 99%

“…Wang and Lyu [43] shown that all Euler sums of weight eight are reducible to linear sums, and proved that all Euler sums of weight nine are reducible to zeta values. In the most recent papers [44,51,52], the first author has carried out fruitful cooperation with Wang proved that all Euler sums of weight ten can be expressed as a rational linear combination of ζ (10) (−1) j−1 j p and Hn := H(1) n , p ∈ N.…”

Section: Harmonic Numbers and Euler Sumsmentioning

confidence: 99%

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On harmonic numbers and nonlinear Euler sums

Cai

2018

Journal of Mathematical Analysis and Applications

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In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight ≤ 5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new.

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Section: Harmonic Numbers and Euler Sumsmentioning

confidence: 99%

Section: Harmonic Numbers and Euler Sumsmentioning

confidence: 99%

On harmonic numbers and nonlinear Euler sums

Cai

2018

Journal of Mathematical Analysis and Applications

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“…From [33], we can find the following relation By the definitions of alternating linear Euler sums and alternating double zeta values, we have S p,q = ζ ⋆ (q, p) , S p,q = −ζ ⋆ (q, p) , S p,q = −ζ ⋆ (q, p) .…”

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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On harmonic numbers and nonlinear Euler sums

Xu¹,

Cai²

2016

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Evaluations of nonlinear Euler sums of weight ten

Cited by 4 publications

References 16 publications

On harmonic numbers and nonlinear Euler sums

On harmonic numbers and nonlinear Euler sums

Some evaluation of cubic Euler sums

On harmonic numbers and nonlinear Euler sums

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