Dirichlet characters of the rational polynomials

Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [ 6 ] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.