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On Euler-Barnes multiple zeta functions. (English) Zbl 1038.11058

Summary: We define the analytic continuation of multiple zeta functions (the Euler-Barnes multiple zeta functions) depending on parameters a 1 ,a 2 ,,a r taking positive values in the complex number field,

ζ r (s,w,ua 1 ,,a r )= m 1 ,,m r =0 u -(m 1 ++m r ) (w+m 1 a 1 ++m r a r ) s ,

where w>0 and u with |u|>1. We also study some interesting properties of the Euler-Barnes multiple zeta functions at negative integers. In the final section, we construct p-adic Euler integrals used in the proof of Witt-type formulas for the Barnes-type multiple Frobenius-Euler numbers.


MSC:
11M41Other Dirichlet series and zeta functions