Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants

Abstract

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connecting obtained formulae with Poincare series of ring of functions on singularities through results of S. M. Gusein-Zade, F. Delgado, and A. Campillo.