In this paper, we first introduce a new second-order non-linear recursive polynomials $U_{h,i}\left(x\right)$ , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums $\mathrm{mod}p$ (a prime with $p\equiv 1\mathrm{mod}12$ ) to give an exact evaluation formula for it.

In this paper, we consider the fourth hybrid power mean involving two-term exponential sums and third-order character sum modulo p, a topic of significant importance in analytic number theory. These results generalize prior research, and provide new insights for studying the relationship between character sums and exponential sums.