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 }, defined by the recurrence P0f(x)=1,P1f(x)=x,Pnf(x)=xn∑k=1nf(k)Pn-kf(x),n≥2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} P_{0}^{f}(x)=1, &{} \\\\ P_{1}^{f}(x)=x, &{} \\\\ P_{n}^{f}(x)=\\frac{x}{n}\\sum _{k=1}^{n}f(k)P_{n-k}^{f}(x), &{} n\\ge 2. \\end{array}\\right. \\end{aligned}$$\\end{document}Using the ideas from the paper by Heim, Luca, and Neuhauser, we prove, under some assumptions on P_{n}^{f}(t) for 1le nle 10, that no root of unity can be a root of any polynomial P_{n}^{f}(t) for nin {mathbb {N}}. Then we specify the result to some functions f related to colored partitions.