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Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 281-288, 2013
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Two identities related to Dirichlet character of polynomials

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Weili Yao, Wenpeng Zhang

* Weili Yao* (corresponding author), Department of Mathematics, College of Sciences, Shanghai University, Shanghai, P. R. China, e-mail: ` yaoweili@shu.edu.cn`; * Wenpeng Zhang*, Department of Mathematics, Northwest University, Xi'an, Shaanxi, P. R. China, e-mail: ` wpzhang@nwu.edu.cn`

**Abstract:** Let $q$ be a positive integer, $\chi$ denote any Dirichlet character $\mod q$. For any integer $m$ with $(m, q)=1$, we define a sum $C(\chi, k, m; q)$ analogous to high-dimensional Kloosterman sums as follows:

C(\chi, k, m; q)=\sumprime_{a_1=1}^q \sumprime_{a_2=1}^q \cdots\sumprime_{a_k=1}^q \chi(a_1+a_2+\cdots+a_k+m\overline{a_1a_2\cdots a_k}),

where $a\cdot\overline{a}\equiv1\bmod q$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $|C(\chi, k, m; q)|$, and give two interesting identities for it.

**Keywords:** Dirichlet character of polynomials, sum analogous to Kloosterman sum, identity, Gauss sum

**Classification (MSC 2010):** 11L05

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