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## Euler Sums and Contour Integral Representations (1998)

Citations: | 44 - 1 self |

### Citations

1303 |
A course of modern analysis
- Whittaker, Watson
- 1927
(Show Context)
Citation Context ...he special residue sum is always determined by a few Taylor series expansions taken at a finite collection of points. We make here an essential use of kernels involving the / function. The / function =-=[15]-=- is the logarithmic derivative of the Gamma function, /(s) = d ds log \Gamma(s) = \Gammafl \Gamma 1 s + 1 X n=1 [ 1 n \Gamma 1 n + s ] (12) and it satisfies the complement formula /(s) \Gamma /(\Gamma... |

283 | Applied and computational complex analysis - Henrici - 1988 |

252 |
Ramanujan’s notebooks, Part I.
- Berndt
- 1985
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Citation Context ...i(2) \Gamma 12 \Delta ln 2 + 45 64 i(4) \Gamma 1 4 i(2) \Gamma 3 32 i(2)i(3) \Gamma 41 8 i(3) + 17 32 i(5): Several related identities but of a simpler form appear in Chap. 9 of Ramanujan's notebooks =-=[2]-=-. This example is typical. A great many evaluations now become of a purely mechanical character. It is then easy to develop general formulae that systematically turn out to be homogeneous convolution ... |

198 |
Multiple harmonic series,
- Hoffman
- 1992
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Citation Context ...CONTOUR INTEGRAL REPRESENTATIONS 3 values [1]. This suggests the existence of both "general" classes of evaluations and "exceptional" evaluations. A recent approach exemplified by =-=the work of Hoffman [9]-=- and Zagier [16] sheds a new light on these phenomena. It is based on considering the multiple zeta functions defined by i(a 1 ; a 2 ; : : : ; a ` ) := X n1!n2!\Delta\Delta\Delta!n ` 1 n a1 1 n a2 2 \... |

108 |
Values of zeta functions and their applications
- Zagier
- 1994
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Citation Context ...L REPRESENTATIONS 3 values [1]. This suggests the existence of both "general" classes of evaluations and "exceptional" evaluations. A recent approach exemplified by the work of Hof=-=fman [9] and Zagier [16]-=- sheds a new light on these phenomena. It is based on considering the multiple zeta functions defined by i(a 1 ; a 2 ; : : : ; a ` ) := X n1!n2!\Delta\Delta\Delta!n ` 1 n a1 1 n a2 2 \Delta \Delta \De... |

64 | Explicit evaluation of Euler sums,
- Borwein, Borwein, et al.
- 1995
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Citation Context ...respond to essentially nontrivial identities of which (a; b; c) of Eq. (3) are typical. Rather extensive numerical search for linear relations between linear Euler sums and polynomials in zeta values =-=[1]-=- strongly suggest that Euler found all the possible evaluations of linear sums. The next object of interest is the nonlinear sums that involve products of at least two harmonic numbers. Let $ = ($ 1 ;... |

47 | An average–case analysis of the Gaussian algorithm for lattice reduction
- Daudé, Flajolet, et al.
- 1997
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Citation Context ...\Gamma 7 4 i(3) log 2; where Li q (z) = P 1 n=1 z n n \Gammaq is the polylogarithm. The constant �� 1 is related to several of Ramanujan 's evaluations as well as to the analysis of lattice reduct=-=ion [6] men-=-tioned in the introduction. Higher order ��'s are not known to be related to classical constants. Following Euler, Nielsen established relations suggesting that alternating sums of odd weight shou... |

47 | Handbuch der Theorie der Gammafunktion. - Teubner, - Nielsen - 1906 |

42 | Ramanujan’s Notebooks. - Berndt - 1991 |

28 |
Triple sums and the Riemann zeta function
- Markett
- 1994
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Citation Context ...u j = 1 \Gamma t j . Partial fraction expansions (\Pi). The Euler-Nielsen method, of which an idea was given at the beginning of Section 3 applies to double zetas [13], and, as established by Markett =-=[12]-=- and Borwein-Girgensohn [5], it can be extended to triple zetas. We let \Pi 2 and \Pi 3 denote the linear relations that derive from this mechanism in the case of zetas of multiplicities 2 and 3. 16 P... |

27 | Hypergeometrics and the cost structure of quadtress,
- Flajolet, Labelle, et al.
- 1995
(Show Context)
Citation Context ... in perturbative quantum field theory. They also surface occasionally in combinatorial mathematics: the Euler evaluation (a) of Eq. (3) serves to analyze the distribution of node degrees in quadtrees =-=[8, 10]-=- while alternating Euler sums make an appearance in the analysis of lattice reduction algorithms [6]. 2. General summations Contour integration is a classical technique for evaluating infinite sums by... |

19 |
Le Calcul des résidus et ses applications à la théorie des fonctions. Gauthier-Villars
- Lindelöf
- 1905
(Show Context)
Citation Context ...e poles at s = 0; \Sigmai yield the explicit form appearing on the right. This summation mechanism is formalized by a lemma that goes back to Cauchy and is nicely developed throughout Lindelof's book =-=[11]. We define a ke-=-rnel function ��(s) by the two requirements:s��(s) is meromorphic in the whole complex plane; ��(s) satisfies ��(s) = o(s) over an infinite collection of circles jzj = ae k with ae k !... |

11 | Evaluation of triple Euler sums,
- Borwein, Girgensohn
- 1996
(Show Context)
Citation Context ...ring all ways of interlacing the vector arguments (a) and (b; c). The conjunction of the theorem and shuffle relations, provides a simple proof of "half " of the main result of Borwein and G=-=irgensohn [5]-=- according to which all triple zeta values of even weight are reducible to double zeta values. The reductions obtained are in addition explicit double convolutions of simple and double zeta values. Co... |

7 |
Combinatorial Variations on Multidimensional Quadtrees
- Labelle, Laforest
- 1995
(Show Context)
Citation Context ... in perturbative quantum field theory. They also surface occasionally in combinatorial mathematics: the Euler evaluation (a) of Eq. (3) serves to analyze the distribution of node degrees in quadtrees =-=[8, 10]-=- while alternating Euler sums make an appearance in the analysis of lattice reduction algorithms [6]. 2. General summations Contour integration is a classical technique for evaluating infinite sums by... |

7 |
A formula of S
- Sitaramachandrarao
- 1987
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Citation Context ...(n) = (\Gamma1) n (H n\Gamma1 \Gamma log 2); /(s) = s!\Gamman (\Gamma1) n 1 (s + n) + (H n \Gamma log 2) + \Delta \Delta \Delta : The following evaluations are all found in Sitaramachandrarao's paper =-=[14]-=- that contains an exhaustive discussion of sums S \Sigma\Sigma 1;r together with a thorough bibliography. Here, the identities come out as simple consequences of the process employed earlier for stand... |

7 | Values of zeta functions and their applications," preprint (Max-Planck Institut). Crandall and Buhler: On the Evaluation of Euler Sums 285 r s Approximate value of i(r; s) 2 0:93182449042503409855161511070364305170750579463468769957662770 0:381330153 - Zagier - 1994 |

5 |
On some series containing /(x) \Gamma /(y) and (/(x) \Gamma /(y)) for certain values of x and y
- Doelder
- 1991
(Show Context)
Citation Context ...;q = S 112225;q = 1 X n=1 (H n ) 2 (H (2) n ) 3 H (5) n n q : In the past, a few basic nonlinear sums have been evaluated thanks to their relations to the Eulerian beta integrals or to polylogarithms =-=[7]-=-. Recently, a detailed numerical search conducted by Bailey, Borwein, Girgensohn [1] has revealed the existence of many surprising evaluations like the cubic or quartic formulae of (3). Some of these ... |

5 | A formula of S - Rao - 1987 |

3 |
notebooks, Part II
- Ramanujan's
- 1989
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Citation Context ...l (�� cot ��s) which, when applied to the functions (�� coth ��s)=s q yields identities like 1 X n=1 coth ��k k 3 = 7 180 �� 3 ; 1 X n=1 coth ��k k 7 = 19 56700 �� 7 th=-=at were discovered by Ramanujan [3]-=-. EULER SUMS AND CONTOUR INTEGRAL REPRESENTATIONS 23 Acknowledgements. Early discussions with Brigitte Vall'ee have helped clarify the residue approach to Euler sums computations. We are grateful to D... |

3 |
Reprinted from Handbuch der Theorie der Gammafunktion
- Nielsen, Gammafunktion, et al.
- 1965
(Show Context)
Citation Context ...nning in 1742 (see [2, p. 253] for a discussion) and he was the first to consider the linear sums, S p;q := 1 X n=1 H (p) n n q : (5) Euler, whose investigations were to be later completed by Nielsen =-=[13]-=-, discovered that the linear sums have evaluations in terms of zeta values in the following cases: p = 1; p = q; p + q odd; p + q even but with the pair (p; q) being restricted to a finite set of so-c... |

2 | Experimental evaluation of Euler sums", Experimental Mathematics 3 - Bailey, Borwein, et al. - 1994 |

1 | Evaluation of triple Euler sums", Electron - Borwein, Girgensohn - 1996 |

1 | On the evaluation of Euler sums", Experiment - Crandall, Buhler - 1994 |

1 | de Doelder, "On some series containing /(x) \Gamma /(y) and (/(x) \Gamma /(y)) for certain values of x and y - J - 1991 |