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

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ 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) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.


Introduction
Let q 3 be a positive integer. For any positive integer k 2, the k-th Gauss sums G.m; kI q/ is defined as where e.y/ D e 2 iy . Recently, some scholars have studied the properties of A.m; kI q/, and obtained many interesting results. For example, Shen Shimeng and Zhang Wenpeng [1] proved a recurrence formula related to A.m; 4I p/. Li Xiaoxue and Hu Jiayuan [2] studied the computational problem of the hybrid power mean where p is an odd prime with p Á 1 mod 4. They proved the identity where 4 denotes any fourth-order character mod p, . / D P p 1 aD1 .a/e a p Á denotes the classical Gauss sums, and c denotes the multiplicative inverse of c mod p.
Li Xiaoxue and Hu Jiayuan [2] also computed the exact value of 2 . 4 / C 2 . 4 / and 5 . 4 / C 5 . 4 /. Other works related to k-th Gauss sums and the generalized k-th Gauss sums can be found in [3][4][5][6][7][8][9][10]. Now let p be an odd prime with p Á 1 mod 4, be any fourth-order character mod p, and we also define G.k; p/ as G.k; p/ D k . / C k : In this paper, we employ a remark of [2] by using the analytic method and the properties of the classical Gauss sums to study the computational problem of G.k; p/ for any positive integer k, and give an interesting fourth-order linear recurrence formula for G.k; p/. That is, we will prove the following result.
Theorem. Let p be an odd prime with p Á 1 mod 4. Then for any positive integer k, we have the linear recurrence formulae From these recurrence formulae, we can obtain the exact value of G.k; p/ for all positive integer k. Hence, we can also deduce the following corollaries: Corollary 1.1. Let p be an odd prime with p Á 5 mod 8. Then we have:

Several Lemmas
In this section, we need several simple lemmas, which are necessary in the proof of our theorem. Hereinafter, we will use many properties of the classical Gauss sums, all of which can be found in [11]; so they will not be repeated here. First we have the following: Lemma 2.1. Let p be an odd prime with p Á 1 mod 4. Then for any quadratic non-residue r mod p, we have the identity where p Á denotes the Legendre's symbol mod p, and a denotes the multiplicative inverse of a mod p.
Proof. This is a well known result. See Theorem 4-11 in [12].
Lemma 2.2. Let p be an odd prime with p Á 1 mod 4, be any fourth-order character mod p. Then we have the identity Proof. Let be any fourth-order character mod p, it is clear that 2 D p Á D 2 , the Legendre's symbol mod p. Then from the definition and properties of the classical Gauss sums we have Similarly, we also have .a/ 2 .a C 1/: As is a fourth-order character mod p, for any integer m with .m; p/ D 1, we have Note that for prime p with p Á 1 mod 4, one has identity . 2 / D p p. From (2) and (3)