Borwein, Jonathan M. and Girgensohn, Roland (1996) *Evaluation of Triple Euler Sums.* The Electronic Journal of Combinatorics, 3 (1).

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## Abstract

Let a,b,c be positive integers and define the so-called triple, double and single Euler sums by ζ(a,b,c) := ∑x=1∞∑y=1x−1∑z=1/y−11x^ay^bz^c, ζ(a,b) := ∑x=1∞∑y=1x−11x^ay^b and ζ(a) := ∑x=1∞1xa. Extending earlier work about double sums, we prove that whenever a+b+c is even or less than 10, then ζ(a,b,c) can be expressed as a rational linear combination of products of double and single Euler sums. The proof involves finding and solving linear equations which relate the different types of sums to each other. We also sketch some applications of these results in Theoretical Physics.

Item Type: | Article |
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Additional Information: | pubdom FALSE |

Uncontrolled Keywords: | Riemann zeta functions, Euler sums, polylogarithms, harmonic numbers, quantum field theory, knot theory |

Subjects: | 40-xx Sequences, series, summability > 40Axx Convergence and divergence of infinite limiting processes 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory 40-xx Sequences, series, summability > 40Bxx Multiple sequences and series 33-xx Special functions > 33Exx Other special functions |

Faculty: | UNSPECIFIED |

Depositing User: | Users 1 not found. |

Date Deposited: | 17 Nov 2003 |

Last Modified: | 13 Sep 2014 21:05 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/118 |

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