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Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth

Published online by Cambridge University Press:  04 December 2007

Douglas Bowman  and
David M. Bradley
Douglas Bowman
Affiliation:
Department of Mathematics, Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, U.S.A. e-mail: bowman@math.niu.edu
David M. Bradley
Affiliation:
Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, ME 04469-5752, U.S.A. e-mail: bradley@math.umaine.edu
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Abstract

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We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.

Keywords

multiple zeta valuesMZVmultiple polylogarithmsEuler sumshypergeometric functions
Type
Research Article
Information
Compositio Mathematica , Volume 139 , Issue 1 , October 2003 , pp. 85 - 100
Copyright
© 2003 Kluwer Academic Publishers