On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

Let ${\mathbb{F}}_q$ be a finite field, ${\mathbb{F}}_{q^s}$ be an extension of ${\mathbb{F}}_q$, let $f(x)\in {\mathbb{F}}_q[x]$ be a polynomial of degree $n$ with $\gcd (n,q)=1$. We present a recursive formula for evaluating the exponential sum ${\sum }_{c\in {\mathbb{F}}_{q^s}}{\chi }^{(s)}(f(x))$. Let $a$ and $b$ be two elements in ${\mathbb{F}}_q$ with $a\mathrm{\ne }0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in $\mathbb{F}$^*_{q^s}}\chi^{(s)}(ac^u+bc^{-1})$,where$\chi^{(s)}$ is the lifting of an additive character $\chi$ of ${\mathbb{F}}_q$. Some properties of the sequences constructed from these exponential sums are provided also.