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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

$L{i}_{{s}_{1},\cdots ,{s}_{k}}\left({z}_{1},\cdots ,{z}_{k}\right)=\sum _{{n}_{1}>\cdots >{n}_{k}>0}\prod _{j=1}^{k}\frac{{z}_{j}^{{n}_{j}}}{{n}_{j}^{{s}_{j}}},$

where ${s}_{1},\cdots ,{s}_{k}$ and ${z}_{1},\cdots ,{z}_{n}$ 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

$\zeta \left({s}_{1},\cdots ,{s}_{k}\right)=\sum _{{n}_{1}>\cdots >{n}_{u}>0}\prod _{j=1}^{k}\frac{1}{{n}_{j}^{{s}_{j}}}·$

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.

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