Jean Bellissard and Stavros Garoufalidis
Abstract: Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic $G$-function (in the sense of Siegel) when the group is free of finite rank. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman's theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a holonomic $G$-function. We ask whether the latter holds for general hyperbolic groups.
Key words: Rational functions, algebraic functions, holonomic functions, $G$-functions, generating series, non-commuting variables, moments, hamiltonians, resolvant, regular languages, context-free languages, Hadamard product, group-ring, free probability, Schur complement method, free group, von Neumann algebras, polynomial Hamiltonians, free probability,spectral theory, norm.
Notes: 7 pages, 0 figures.