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Values of twisted Barnes zeta functions at negative integers

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Abstract

In this paper, we study analytical and arithmetical properties of the twisted zeta function \(\Gamma (s)^{ - 1} \int_0^\infty {e^{ - xt} t^{s - 1} } \prod\nolimits_{j = 1}^N {\frac{{a_j t - \log (w^a j)}} {{1 - w^{a_j } e^{a_j t} }}dt} \), where ℜ(s) > N, ℜ(x) > 0, w ∈ ℂ\{0}, N ∈ ℤ, and a 1, …, a N have positive real parts. These functions have many interesting properties. We prove a collection of fundamental identities satisfied by zeta functions of this kind. For instance, special values of these zeta functions are related to twisted Barnes numbers and polynomials. This gives us a new elementary approach to new and known results concerning the Barnes zeta functions. In particular, we derive some well-known results on the Hurwitz zeta functions.

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Authors and Affiliations

  1. Département de mathématiques, Université d’Evry Val d’Essonne, Bâtiment I.B.G.B.I., 3ème étage, 23 Bd. de France, 91037, Evry Cedex, France

    A. Bayad

  2. Faculty of Arts and Science, Department of Mathematics, University of Akdeniz, 07058, Antalya, Turkey

    Y. Simsek

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  1. A. Bayad
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  2. Y. Simsek
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Correspondence to A. Bayad.

Additional information

This work was supported by Akdeniz University Scientific Research Projects Unit and Univeristé Val d’Essonne Scientific Research Programs.

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Bayad, A., Simsek, Y. Values of twisted Barnes zeta functions at negative integers. Russ. J. Math. Phys. 20, 129–137 (2013). https://doi.org/10.1134/S1061920813020015

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  • DOI: https://doi.org/10.1134/S1061920813020015

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