Estimate for the largest zeros of the D’Arcais polynomials

The zeros of the nth D’Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind \(\eta \)-function. In this paper, we prove that these coefficients are non-vanishing for \(\vert x \vert > \kappa \, (n-1)\) and \(\kappa \approx 9.7225\). Numerical computations imply that 9.72245 is a lower bound for \(\kappa \). This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions.