Estimate for the largest zeros of the D’Arcais polynomials

Abstract

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 $$\eta $$ η -function. In this paper, we prove that these coefficients are non-vanishing for $$\vert x \vert > \kappa \, (n-1)$$ | x | > κ ( n - 1 ) and $$\kappa \approx 9.7225$$ κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for $$\kappa $$ κ . 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.

Raising Series to A Power

“…The class of polynomials τ α (n) are called D' Arcais polynomials (refer (4) ). The search for reducibility criterions of these polynomials over the ring of integers is of special interest and one can see a lot of papers appearing in this direction (refer (2,(5)(6)(7) ), the main reason behind this search is that the non-vanishing of the polynomials at α = −24 for each n is equivalent to Lehmer's conjecture on Ramanujan's tau function which is still open. As the relations (3) and ( 4) are well-known we take the above derivation as an illustration, and proceed in similar fashion taking into account the other partition-generating functions.…”