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- W3103445057 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 equation [ operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}operatorname{Hilb}_A(t), ] where $d$ 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 $operatorname{Hilb}_A(t)$ at $t=1$. The main idea consists of examining the graded algebra $mathcal F=bigoplus_{rin mathbb{Z}}mathcal F_r$ of formal power series in the variable $x$ that fulfill the condition $varphi(x/(x-1))=(1-x)^rvarphi(x)$. 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." @default.
- W3103445057 created "2020-11-23" @default.
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- W3103445057 date "2019-02-09" @default.
- W3103445057 modified "2023-10-01" @default.
- W3103445057 title "The Laurent Coefficients of the Hilbert Series of a Gorenstein Algebra" @default.
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- W3103445057 doi "https://doi.org/10.1080/10586458.2018.1492473" @default.
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