  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.
