On the fourth-order linear recurrence formula related to classical Gauss sums

Let p be an odd prime with p equivalent to 1 mod 4, k be any positive integer, psi be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G (k, p) = tau(k) (psi) + tau(k) ((psi) over bar) and give an interesting fourth-order linear recurrence formula for it, where tau(psi) denotes the classical Gauss sums.