The Laurent Coefficients of the Hilbert Series of a Gorenstein Algebra

Abstract

Dedicated to the memory of Leslie Tanner Hammontree.Abstract. By a theorem of R. Stanley, a graded Cohen-Macaulay domain A is Gorenstein if and only if its Hilbert series satisfies the functional equationwhere d is the Krull dimension and a is the a-invariant of A. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of Hilb A (t) at t = 1. The main idea consists of examining the graded algebra F = r∈Z Fr of formal power series in the variable x that fulfill the condition ϕ(x/(x − 1)) = (1 − x) r ϕ(x). As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree r = −(a + d) = 0, these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.