The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann-Manschot and about the asymptotic behavior of the chi(y)-genera of Hilbert schemes of points on K3 surfaces by Manschot-Rolon. We also get the algebraicity of the generating functions given by Gottsche for the Hilbert schemes associated to general algebraic surfaces. (C) 2018 Elsevier Inc. All rights reserved.