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Euler sums and contour integral representations. (English) Zbl 0920.11061
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.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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References:
[1] Bailey D. H., Experiment. Math. 3 (1) pp 17– (1994) · Zbl 0810.11076
[2] Berndt B. C., Ramanujan’s notebooks (1985) · Zbl 0555.10001
[3] Berndt B. C., Ramanujan’s notebooks (1989) · Zbl 0716.11001
[4] Borwein J. M., Electron. J. Combin. 3 (1) pp 27– (1996)
[5] Borwein D., Proc. Edinburgh Math. Soc. 38 (2) pp 2– (1995)
[6] Crandall R. E., Experiment. Math. 3 (4) pp 275– (1994) · Zbl 0833.11045
[7] Daudé H., Combinatorics, Probability, and Computing 6 (4) pp 397– (1997) · Zbl 0921.11072
[8] de Doelder R. J., J. Comput. Appl. Math. 37 pp 1– (1991)
[9] Flajolet R, Random Structures Algorithms 7 (2) pp 117– (1995) · Zbl 0834.68013
[10] Henrici P., Applied and computational complex analysis, Pure and Applied Mathematics (1974) · Zbl 0313.30001
[11] Hoffinan M. E., Pacific J. Math. 152 (2) pp 275– (1992)
[12] Labelle G., J. Combin. Theory Ser. A 69 (1) pp 1– (1995) · Zbl 0815.05002
[13] Lindelöf E., Le calcul des résidus et ses applications á la théorie des fonctions (1905) · JFM 36.0468.01
[14] Markett C., J. Number Theory 48 (2) pp 113– (1994) · Zbl 0810.11047
[15] Nielsen N., Handbuch der Theorie der Gammafunktion and Theorie des Integrallogarithmus und verwandter Transzendenten (1906)
[16] Sitaramachandra Rao R., J. Number Theory 25 (1) pp 1– (1987) · Zbl 0606.10032
[17] Whittaker E. T., A course of modern analysis,, 4. ed. (1927)
[18] Zagier D., First European Congress of Mathematics pp 497– (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.