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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 $a_1,a_2,\dots,a_r$ taking positive values in the complex number field, $$\zeta_r(s,w,u\mid a_1,\dots,a_r)=\sum^\infty_{m_1,\dots,m_r=0}\ \frac{u^{-(m_1+\cdots+m_r)}}{(w+m_1a_1+\cdots+m_ra_r)^s}\,,$$ where $\Re w > 0$ and $u\in \bbfC$ with $\vert u\vert > 1$. 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.

11M41Other Dirichlet series and zeta functions