# Is there a name for these kinds of polynomials?

I've come across the following polynomials in my research and I am wondering if they have a name or if there is very much known about them:

$$\begin{equation} F_{\chi}(T) = \sum_{a = 1}^{n-1} \chi(a)T^a \end{equation}$$

where $$\chi$$ is a Dirichlet character of conductor $$n$$. The closest thing I can find on these are the Fekete polynomials (this is the special case where $$\chi$$ is the Legendre symbol and $$n$$ is a prime number) but nothing for general Dirichlet characters. Thanks in advance for any information you might be able give me.

• Good morning! There are several keywords that may apply here. If $\chi$ is a Dirichlet character mod. $p$ of order $k$ and $T=e^{\frac{2\pi i}{p}}$, you have a "Gauss sum" (cf. projecteuclid.org/download/pdf_1/euclid.bams/1183548292); I believe that in such a case, the phrase "hybrid character sum" applies, too. If $T$ is not of the aforementioned form, an expression that may apply to your $F_{\chi}(T)$ is "twisted character sum". Dec 29, 2020 at 8:36
• Don't know it this helps, but these polynomials appear in the definiton of generalized Bernoulli numbers and polynomials.See for example Lang's "Cyclotomic Fields (vols I and II combined)", page 37.
– efs
Jan 10, 2021 at 23:50