In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and \(0<h(n) \le h(n+1)\). We put \(P_0^{g,h}(x)=1\) and

As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind \(\eta \)-function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

在本文中，我们研究了算术函数g和h的增长特性和多项式的零分布，其中g是中等增长的归一化，并且\(0<h(n) \le h(n+1)\)。我们把\(P_0^{g,h}(x)=1\)和

作为一个应用，我们在 Dedekind \(\eta \)函数的幂的傅里叶系数不为零域上获得了最著名的结果。这里，g是除数之和，h 是恒等函数。 Kostant 关于简单复李代数表示的结果和 Han 关于 Nekrasov-Okounkov 钩长度公式的结果得到了扩展。这些多项式与爱森斯坦级数、克莱因j不变量和第二类切比雪夫多项式的倒数相关。