Unusual class of polynomials related to partitions

Abstract In this paper, we study, for a given arithmetic function f , the sequence of polynomials $$(P_{n}^{f}(t))_{n=0}^{\infty }$$ ( P n f t ) = 0 ∞ defined by recurrence $$\begin{aligned} \left\{ \begin{array}{ll} P_{0}^{f}(x)=1, &{} \\ P_{1}^{f}(x)=x, P_{n}^{f}(x)=\frac{x}{n}\sum _{k=1}^{n}f(k)P_{n-k}^{f}(x), n\ge 2. \end{array}\right. \end{aligned}$$ x 1 , ∑ k - ≥ 2 . Using ideas from paper Heim, Luca, and Neuhauser, prove, under some assumptions on $$P_{n}^{f}(t)$$ $$1\le n\le 10$$ ≤ 10 that no root unity can be any polynomial $$n\in {\mathbb {N}}$$ ∈ N . Then specify result to functions related colored partitions.