Estimate for the largest zeros of the D’Arcais polynomials

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-15 DOI:10.1007/s40687-023-00412-z

Bernhard Heim, Markus Neuhauser

引用次数: 0

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 \(\eta \)-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.

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估计D 'Arcais多项式的最大零

第n个D 'Arcais多项式(在组合学中也称为Nekrasov-Okounkov多项式)的零点决定了Dedekind \(\eta \) -函数的所有复幂x的第n个傅立叶系数的消失性质。在本文中，我们证明了这些系数对于\(\vert x \vert > \kappa \, (n-1)\)和\(\kappa \approx 9.7225\)是不消失的。数值计算表明，9.72245是\(\kappa \)的下界。这大大改进了Kostant、Han和Heim-Neuhauser先前的结果。本文研究的多项式包括第二类Chebyshev多项式、1相关的Laguerre多项式、Hermite多项式、过分割多项式和平面分割多项式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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