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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 $Li_{s_1, \dots, s_k} (z_1, \dots,z_k)= \sum_{n_1> \cdots> n_k>0} \prod^k_{j=1} {z_j^{n_j}\over n_j^{ s_j}},$ where $$s_1,\dots, s_k$$ and $$z_1,\dots,z_n$$ are complex numbers suitably restricted so that the sum converges. When $$z_1=\cdots =z_k=1$$, the multiple polylogarithm reduces to a multiple zeta function $\zeta(s_1, \dots,s_k)= \sum_{n_1> \cdots>n_u >0}\prod^k_{j=1} {1\over n_j^{s_j}}.$ 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.
For the entire collection see [Zbl 0980.00024].

##### MSC:
 33E20 Other functions defined by series and integrals 11G55 Polylogarithms and relations with $$K$$-theory 40B05 Multiple sequences and series