LAUSR

Let eq be a nontrivial additive character of a finite field ????q of order q ≡ 1(mod 3) and let ψ be a cubic multiplicative character of ????q, ψ(0) = 0. Consider the cubic Gauss sum and the cubic exponential sumGψ=∑z∈Fqeqzψz,Cω=∑z∈Fqeqz3ω−3z,ω∈Fq,ω≠0.$$ G\left(\psi \right)=\sum \limits_{z\in {\mathbb{F}}_q}{e}_q(z)\psi (z),\kern0.5em C\left(\omega \right)=\sum \limits_{z\in {\mathbb{F}}_q}{e}_q\left(\frac{z^3}{\omega }-3z\right),\kern0.5em \omega \in {\mathbb{F}}_q,\kern1em \omega \ne 0. $$It is proved that for all nonzero a, b ∈ ????q,1q∑nCanCbnψn+1qψabGψ2=ψ¯abψa−bGψ¯,$$ \frac{1}{q}\sum \limits_nC(an)C(bn)\psi (n)+\frac{1}{q}\psi (ab)G{\left(\psi \right)}^2=\overline{\psi}(ab)\psi \left(a-b\right)\overline{G\left(\psi \right)}, $$where the summation runs over all nonzero n ∈ ????q.