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)(infinity), defined by the recurrence

Using the ideas from the paper by Heim, Luca, and Neuhauser, we prove, under some assumptions on P-n(f) (t) for 1 <= n <= 10, that no root of unity can be a root of any polynomial P-n(f) (t) for n is an element of N. Then we specify the result to some functions f related to colored partitions.

Keywords

• Partition polynomials
• Colored partitions
• Roots of unity
• Sum of divisor function