On the growth and zeros of polynomials attached to arithmetic functions

Abstract

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 $0<h(n) \leq h(n+1)$. We put $P_0^{g,h}(x)=1$ and \begin{equation*} P_n^{g,h}(x) := \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{equation*} 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.