Hoang Ngoc Minh; Jacob, Gérad; Petitot, Michel; Oussous, Nour Eddine Combinatorial aspects of polylogarithms and Euler-Zagier sums. (Aspects combinatoires des polylogarithmes et des sommes d’Euler-Zagier.) (French) Zbl 0964.33003 Sémin. Lothar. Comb. 43, B43e, 29 p. (1999). Summary: The algebra of polylogarithms is the smallest \(\mathbb{C}\)-algebra which contains the constants and which is stable under integration with respect to the differential forms \(dz/z\) and \(dz/(1-z)\). It is known that this algebra is isomorphic to the algebra of the noncommutative polynomials equipped with the shuffle product. As a consequence, the polylogarithms \(\text{Li}_n(g(z))\) with \(n\geq 1\), where the \(g(z)\) belong to the group of biratios, are the polylogarithms indexed by Lyndon words with coefficients in a certain transcendental extension of \(\mathbb{Q}\): the algebra of the Euler-Zagier sums. We conjecture that this algebra is an algebra of polynomials, and we attempt to find a basis for this algebra. The question of knowing whether the polylogarithms \(\text{Li}_n(g(z))\) satisfy a linear functional equation is effectively decidable up to a conjecture of Zagier about the dimension of the algebra. This decision procedure makes use of the decomposition of those polylogarithms indexed by the Lyndon basis. Such an algorithm is based on the factorisation of the generating function of these polylogarithms. Cited in 15 Documents MSC: 33B30 Higher logarithm functions 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 05E99 Algebraic combinatorics Keywords:polylogarithms × Cite Format Result Cite Review PDF Full Text: EuDML EMIS