Abstract
AbstractIn 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
)
)
n
=
0
∞
, defined by the recurrence $$\begin{aligned} \left\{ \begin{array}{ll} P_{0}^{f}(x)=1, &{} \\ P_{1}^{f}(x)=x, &{} \\ P_{n}^{f}(x)=\frac{x}{n}\sum _{k=1}^{n}f(k)P_{n-k}^{f}(x), &{} n\ge 2. \end{array}\right. \end{aligned}$$
P
0
f
(
x
)
=
1
,
P
1
f
(
x
)
=
x
,
P
n
f
(
x
)
=
x
n
∑
k
=
1
n
f
(
k
)
P
n
-
k
f
(
x
)
,
n
≥
2
.
Using the ideas from the paper by Heim, Luca, and Neuhauser, we prove, under some assumptions on $$P_{n}^{f}(t)$$
P
n
f
(
t
)
for $$1\le n\le 10$$
1
≤
n
≤
10
, that no root of unity can be a root of any polynomial $$P_{n}^{f}(t)$$
P
n
f
(
t
)
for $$n\in {\mathbb {N}}$$
n
∈
N
. Then we specify the result to some functions f related to colored partitions.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference10 articles.
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5. Heim, B., Luca, F., Neuhauser, M.: On cyclotomic factors of polynomials related to modular forms. Ramanujan J. 48, 445–458 (2019)