Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) = \sum_{x \in \mathbb{F}_q^*} \psi\left(ax + \dfrac{b}{x} \right) $$ which is well-known to satisfy $|K(a,b)| \leq 2q^{1/2}$.
I wonder if anyone has seen the following natural "generalization", and if so, what is an upper bound for its modulus?
Let $m \geq 1$ and define $$ K_m(a,b) = \sum_{x \in \mathbb{F}_q^*} \psi\left(a\left(x^m + x^{m-1} + \cdots + x \right) + b\left(\dfrac{1}{x^m} + \dfrac{1}{x^{m-1}} + \cdots + \dfrac{1}{x} \right)\right). $$ Note that $K_1(a,b) = K(a,b)$. In the classical $K(a,b)$, note the 2 in the upper bound for the modulus, which (maybe naive to think so!) may have to do with the fact that the argument $ax + \dfrac{b}{x} = (ax^2 + b)/x$. Thus perhaps $|K_m(a,b)| \leq 2mq^{1/2}$? Have you guys seen this kind of sum before? Thanks!
Yes. Bounds for such sums are known more generally for Laurent polynomials. It is a useful (but lengthy) exercise to derive the bound $$ \left|\sum_{x\in\Bbb{F}_q^*}\psi( f(x)+g(\frac1x))\right|\le (\deg f+\deg g)\sqrt q $$ with the method described in Lidl & Niederreiter. Here $f$ and $g$ can be any polynomials that cannot be written in the form $h^p-h+c$ for some other polynomial $h$ and constant $c$.
But I'm afraid I cannot point you at a definite source. In the 90s I was among a group of coding theorists who desperately needed these bounds for certain constructions. We also needed related hybrid sums, where you throw a multiplicative character $\chi$ into the mix: $$ \left|\sum_{x\in\Bbb{F}_q^*}\chi(x)\psi( f(x)+g(\frac1x))\right|\le (\deg f+\deg g)\sqrt q $$ For lack of a definite reference we needed to (re)derive these bounds by turning the crank following the elementary arguments in L&N (and earlier Wolfgang Schmidt's exposition on the Schmidt-Stepanov method). We, most notably Helleseth, Kumar and Shanbhag, extended these results, by turning the crank a bit more, for Galois rings and their characters.
When we consulted the number theorists, they told us that all those results follow from class field -theoretic arguments (conductors and such), but the references they gave did not involve the function fields. Things have improved a bit since, below I list some sources I found useful.
The class field theoretic terms have been carefully edited away from those expositions, but I recall learning a little about the connections from studying Michael Rosen's book. IIRC I identified ray class groups in action in the argument from one of the examples in Rosen's book. I also want to mention books by Wen-Ching Li and Henning Stichtenoth. The latter has an IMO most accessible account of Stepanov's elementary proof of the Riemann hypothesis for function fields - following Bombieri's idea of writing it in the language of divisors.
Hope this helps.