Some remarks on Hilbert functions of veronese algebras

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R, t) = a(t)/(1 - t(l))(n) for the Poincare series of R as a rational function, we study for 0 less than or equal to i less than or equal to l the graded subspaces circle plus(k)R(kl+i) (which we denote R[l; i]) of R, in particular their Poincare series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zero R[l; i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.