Flajolet, Philippe; Salvy, Bruno Euler sums and contour integral representations. (English) Zbl 0920.11061 Exp. Math. 7, No. 1, 15-35 (1998). 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. Reviewer: Tom M.Apostol (Pasadena) Cited in 12 ReviewsCited in 154 Documents MSC: 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:contour integral representations; survey; Euler sums; alternating Euler sums × Cite Format Result Cite Review PDF Full Text: DOI EuDML EMIS Link Digital Library of Mathematical Functions: In §25.16(ii) Euler Sums ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions §25.16(ii) Euler Sums ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions §25.16(ii) Euler Sums ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions Online Encyclopedia of Integer Sequences: a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041). Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 0. Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)). Ranks of certain relations among Euler sums of weight n. Ranks of certain relations among Euler sums of weight n. Ranks of certain relations among Euler sums of weight n. Ranks of certain relations among Euler sums of weight n. a(n) = 2^(n-5) - A000931(n). Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number. Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) H(n,2)/n^4 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2. Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3. Decimal expansion of sum_(n>=1) H(n)^2/n^5 where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) H(n)^3/n^4 where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)). Decimal expansion of sum_(n>=1) H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)). Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)). Decimal expansion of sum_(n>=1) H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) (H(n,2)/n^2) where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2. Decimal expansion of sum_(n>=1) (H(n,3)/n^3) where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3. Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^2) where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^3) where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^3) where H(n) is the n-th harmonic number. Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^6) where H(n) is the n-th harmonic number. Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)*(2n)*(2n+1))^3, where H(n) is the n-th harmonic number. Decimal expansion of Sum_{n >= 0} (-1)^n*H(n)/(2n+1)^3, where H(n) is the n-th harmonic number. Decimal expansion of Sum_{n >= 1} coth(Pi*n)/n^7 = (19/56700)*Pi^7. References: [1] Bailey D. H., Experiment. Math. 3 (1) pp 17– (1994) · Zbl 0810.11076 · doi:10.1080/10586458.1994.10504573 [2] Berndt B. C., Ramanujan’s notebooks (1985) · Zbl 0555.10001 · doi:10.1007/978-1-4612-1088-7 [3] Berndt B. C., Ramanujan’s notebooks (1989) · Zbl 0716.11001 · doi:10.1007/978-1-4612-4530-8 [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. 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