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Zbl 0920.11061
Flajolet, Philippe; Salvy, Bruno
Euler sums and contour integral representations. (English)
[J] Exp. Math. 7, No.1, 15-35 (1998). ISSN 1058-6458; ISSN 1944-950X/e

The authors survey some of the methods that have been used to study Euler sums, and they introduce a powerful new approach. They apply residue calculus to integrals of the form $$\int_{(\infty)}r(s)\xi(s) ds,$$ where $\int_{(\infty)}$ is the limit of integrals taken along large circles that expand to $\infty$, $r(s)$ is a rational function that is $O(s^{-2})$ for large $| s|$, and $\xi(s)$ is a kernel function that is $o(s)$ on large circles whose radii tend to $\infty$. By employing kernels that are polynomials in $\psi(s)=\Gamma'(s)/\Gamma(s)$, its derivatives and related trigonometric functions, they deduce a host of known relations on Euler sums and discover many new ones. A modification also gives results on alternating Euler sums.
[Tom M.Apostol (Pasadena)]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: contour integral representations; survey; Euler sums; alternating Euler sums

Cited in: Zbl 1234.11022 Zbl 1198.05010 Zbl 1097.11044

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