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Special values of multiple polylogarithms. (English) Zbl 1002.11093
This long and elegant paper would have delighted Euler, and he might even have been surprised at the comprehensive generalizations of some of his better known evaluations. The multiple polylogarithm studied here is defined by \[ \lambda \binom {s_1,\dots, s_k}{b_1,\dots, b_k}:= \sum_{\nu_1,\dots, \nu_k=1}^\infty \prod_{j=1}^k b_j^{-\nu_j} \Biggl( \sum_{i=j}^k \nu_i\Biggr)^{-s_j}, \] which reduces to the classical polylogarithm when \(k=1\), and to the Riemann zeta-function when also \(b=1\). Using the multiple polylogarithm, a number of previously isolated results can be brought together into a coherent framework, as well as providing proofs of several previously conjectured evaluations. In recognition of the Riemann \(\zeta\)-function connection, the multiple polylogarithm in the case where \(b_j=1\) \((j=1,\dots, k)\) is denoted by \(\zeta(s_1,\dots, s_k):= \lambda\binom {s_1,\dots, s_k} {1,\dots, 1}\), and the symbol \(\{s_1,\dots, s_r\}^n\) denotes the string \(s_1,\dots, s_r\) repeated \(n\) times. Thus, for example, the \[ \zeta(\{2\}^n):= \sum_{\nu_1,\dots, \nu_n=1}^\infty \prod_{j=1}^n (\nu_j+\dots +\nu_n)^{-2}= \pi^{2n}/ (2n+1)! \] generalizes Euler’s famous result \(\zeta(2):= \sum_{\nu=1}^\infty \nu^2= \pi^2/6\), while a previous conjecture of Don Zagier, that \[ \zeta(\{3,1\}^n)= 4^{-n} \zeta(\{4\}^n)= 2\pi^{4n}/ (4n+2)!, \] is proved as a corollary of Theorem 11.1, in which a generating function involving polylogarithm coefficients is evaluated as a product of two \({}_2F_1\) hypergeometric functions. The result \(\zeta(\{3,1\}^n)= 4^{-n} \zeta(\{4\}^n)\) just quoted, is an example of a “multiple zeta value” (MZV) reduction, and §3 of the paper is devoted to studying more general MZV reductions. Another section (§4) deals with integral representations involving polylogarithms, while §5 concerns “shuffles and stuffles”, studying the combinatorics of the behaviour of multiple polylogarithms with respect to their argument strings.
Indeed, it is difficult in a short review to give adequate expressions to the wealth of ideas presented: this is a fascinating paper which is enhanced by the very comprehensive bibliography of 69 items relating to the subject.

MSC:
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11Y60 Evaluation of number-theoretic constants
11G55 Polylogarithms and relations with \(K\)-theory
33B30 Higher logarithm functions
05A19 Combinatorial identities, bijective combinatorics
33E20 Other functions defined by series and integrals
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