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 h , where is normalized, moderate growth, $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ + 1 . We put $$P_0^{g,h}(x)=1$$ P g , x = $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ : ∑ k - . As an application obtain best known result on domain non-vanishing Fourier coefficients powers Dedekind $$\eta $$ η -function. Here, sum divisors identity function. Kostant’s representation simple complex Lie algebras Han’s results Nekrasov–Okounkov hook length formula are extended. The related reciprocals Eisenstein series, Klein’s j -invariant, Chebyshev second kind.