In this paper, the fourth power mean values of the generalized quadratic Gauss sums associated with the $3$-order and $4$-order Dirichlet characters are given by using the properties of the Dirichlet characters and Gauss sums.

This paper mainly discussed the power sums of bi-periodic Fibonacci and Lucas polynomials. In addition, we generalized these results to obtain several congruences involving the divisible properties of bi-periodic Fibonacci and Lucas polynomials.