In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two-term exponential sums and the cubic Gauss sums. As an application of this result, some exact computational formulas for one kind hybrid power mean of trigonometric sums are obtained.

This paper establishes symmetric inequalities for the reciprocal sums of Fibonacci numbers. By using elementary methods, recurrence properties, Cassini’s identity, and the Fibonacci–Lucas connection, we deduce the precise bounds for these reciprocal sums and an asymptotic formula for the corresponding floor function. This work is novel in two key aspects: it is the first to combine reciprocals of linear recurrence sequences with the tail of the Riemann zeta function, and the first to derive symmetric inequalities for such reciprocal sums.

In this paper, we investigate the fourth moments of two-term exponential sums weighted by Dirichlet characters. Using properties of quadratic residues and the third-order character \( \psi \) modulo a prime \( p \), we first present an explicit identity for the fourth moment of cubic exponential sums weighted by the Legendre symbol when \( (3,p-1)=3 \). Subsequently, by counting solutions to certain congruence equations, we derive some results on the fourth mixed moments associated with quartic exponential sums weighted by an arbitrary Dirichlet character \( \chi\bmod p \).

In this paper, we use the elementary and analytic methods to investigate the computational problem of quadratic character sums related to quaternary and quintuple symmetric polynomials modulo a prime $ p $, respectively. By applying properties of Gauss sums and those of third-order characters modulo $ p $, we establish some identities for such quadratic character sums in the cases of $ (3, p-1) = 1 $ and $ (3, p-1) = 3 $.