Ramanujan J. 46:3 (2018), 633-655; arXiv:1610.07793
Mathematics Subject Classification (2000): Primary 05A17, 14C05, 14G10, 14N10, Secondary 05A30, 11P84, 14G15
Abstract. We compute the coefficients of the polynomials Cn(q) defined by the equation \begin{equation*} 1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{equation*} As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of n points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials Cn(q) are divisible by (q-1)2. We also compute the coefficients of the polynomials P_n(q) = Cn(q)/(q-1)2: each coefficient counts the divisors of n in a certain interval; it is thus a non-negative integer. Finally we give arithmetical interpretations for the values of Cn(q) and of Pn(q) at q = -1 and at roots of unity of order 3, 4, 6.
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(5 octobre 2018)