Weil sums are exponential sums of the form \(\sum_{x\in \mathbb{F}_q} \chi(f(x))\), where \(\mathbb{F}_q\) is the finite field of \(q=p^e\) elements \((p\) prime) and \(f(x)\in\mathbb{F}_q(x)\). In a recent paper [Explicit evaluation of some Weil sums, Acta Arith. 83, 241-251 (1998)] the author gave an explicit evaluation of all Weil sums with \(f(x)=ax^{p^\alpha+1}\) \((p\) odd). The method used was a generalization of Carlitz’ method of a 1980 paper where Weil sums with \(f(x)=ax^{p+1}+bx\) were evaluated.
In the present paper, the author explicitly evaluates sums of the above form with \(f(x)=ax^{p^\alpha+1} +bx \) \((b\neq 0)\). Some related results, and extensions to Weil sums with \(f(x)= ax^{p^\alpha+1}+L(x)\), \(L\) a linearized polynomial, are also included.