A hybrid mean value involving two-term exponential sums and polynomial character sums

Let q ? 3 be a positive integer. For any integers m and n, the two-term exponential sum C(m, n, k; q) is defined by \(C(m,n,k;q) = \sum\limits_{a = 1}^q {e((ma^k + na)/q)} \) , where \(e(y) = e^{2\pi iy} \) . In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.