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Laurent series expansions of multiple zeta-functions of Euler–Zagier type at integer points

Laurent series expansions of multiple zeta-functions of Euler–Zagier type at integer points We give explicit expressions (or at least an algorithm to obtain such expressions) of the coefficients of the Laurent series expansions of the Euler–Zagier multiple zeta-functions at any integer points. The main tools are the Mellin–Barnes integral formula and the harmonic product formulas. The Mellin–Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence. 1 Introduction and the statement of main results The Euler–Zagier r-uple zeta-function is defined by (Hoffman [5], Zagier [15]) ∞ ∞ ∞ −s −s −s 1 2 r ζ (s) = ··· n (n + n ) ··· (n + ··· + n ) (1.1) r 1 2 1 r n =1 n =1 n =1 1 2 r (where s = (s ,..., s ) ∈ C ) in the domain of its absolute convergence, which is 1 r D ={s ∈ C |(s( j , r )) > r − j + 1 (1 ≤ j ≤ r )}, (1.2) This research was partially supported by Grants-in-Aid for Scientific Research, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Laurent series expansions of multiple zeta-functions of Euler–Zagier type at integer points

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References (15)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-019-02337-2
Publisher site
See Article on Publisher Site

Abstract

We give explicit expressions (or at least an algorithm to obtain such expressions) of the coefficients of the Laurent series expansions of the Euler–Zagier multiple zeta-functions at any integer points. The main tools are the Mellin–Barnes integral formula and the harmonic product formulas. The Mellin–Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence. 1 Introduction and the statement of main results The Euler–Zagier r-uple zeta-function is defined by (Hoffman [5], Zagier [15]) ∞ ∞ ∞ −s −s −s 1 2 r ζ (s) = ··· n (n + n ) ··· (n + ··· + n ) (1.1) r 1 2 1 r n =1 n =1 n =1 1 2 r (where s = (s ,..., s ) ∈ C ) in the domain of its absolute convergence, which is 1 r D ={s ∈ C |(s( j , r )) > r − j + 1 (1 ≤ j ≤ r )}, (1.2) This research was partially supported by Grants-in-Aid for Scientific Research,

Journal

Mathematische Zeitschrift – Springer Journals

Published: Jul 5, 2019

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