Formulas for coefficients of polynomials assigned to arithmetic functions

Abstract

We attach to normalized (non-vanishing) arithmetic functions $g$ and $h$ recursively defined polynomials. Let $P_0^{g,h}(x):=1$. Then \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} For special $g$ and $h$, we obtain the D'Arcais polynomials, which are equal to the coefficients of the $-z$th powers of the Dedekind $\eta$-function and are also given by Nekrasov and Okounkov as a hook length formula. Examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and associated Laguerre polynomials. We present explicit formulas and identities for the coefficients of $P_n^{g,h}(x)$ which separate the impact of $g$ and $h$. Finally, we provide several applications.