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Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula

  Published:2010-12-29
 Printed: Apr 2011
  • Driss Essouabri,
    PRES Université de Lyon, Université Jean-Monnet (Saint-Etienne), Faculté des Sciences, Département de Mathématiques, 42023 Saint-Etienne Cedex 2, France
  • Kohji Matsumoto,
    Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
  • Hirofumi Tsumura,
    Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397 Japan
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Abstract

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.
Keywords: Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas
MSC Classifications: 11M41, 40B05, 11B39 show english descriptions Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Multiple sequences and series (should also be assigned at least one other classification number in this section)
Fibonacci and Lucas numbers and polynomials and generalizations
11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
40B05 - Multiple sequences and series (should also be assigned at least one other classification number in this section)
11B39 - Fibonacci and Lucas numbers and polynomials and generalizations
 

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