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 $a\neq 0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum{c\in \mathbb{F}^{*_{qs}}\chi{(s)}(acu+bc{-1})$,} 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.