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Multiple polylogarithms: A brief survey. (English) Zbl 0998.33013
Berndt, Bruce C. (ed.) et al., q-series with applications to combinatorics, number theory, and physics. Proceedings of a conference, University of Illinois, Urbana-Champaign, IL, USA, October 26-28, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 291, 71-92 (2001).

The multiple polylogarithm is defined by

Li s 1 ,,s k (z 1 ,,z k )= n 1 >>n k >0 j=1 k z j n j n j s j ,

where s 1 ,,s k and z 1 ,,z n are complex numbers suitably restricted so that the sum converges. When z 1 ==z k =1, the multiple polylogarithm reduces to a multiple zeta function

ζ(s 1 ,,s k )= n 1 >>n u >0 j=1 k 1 n j s j ·

As the title of the paper indicates, a survey of many results and conjectures for multiple polylogarithms and multiple zeta functions is given. Generally, proofs do not appear here, but an extensive list of references is provided. A new integral representation for the multiple polylogarithm is given, and a q-analogue of the shuffle product is developed.

33E20Functions defined by series and integrals
11G55Polylogarithms and relations with K-theory
40B05Multiple sequences and series