On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums

The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime.


Introduction
where c denotes the multiplicative inverse of c mod p. That is, c ⋅ c ≡ mod p.
For p ≡ mod , they used the elementary method to obtain an interesting third-order linear recurrence formula for S k (p).
Li Xiaoxue and Hu Jiayuan [2] studied the computational problem of the hybrid power mean and proved an exact computational formula for (1).
In this paper, we will consider the calculating problem of the following hybrid power mean: where k ≥ is an integer. If p = h + , then from the properties of the Legendre's symbol mod p we have (see [14], formula (30) in where χ = * p denotes the Legendre's symbol mod p. So in this case, the problem we considered in (2) is trivial. If p = h + , then the situation is more complicated. We will use the analytic method and the properties of classical Gauss sums to study this problem, and prove some new interesting fourth-order linear recurrence formulae for (2) with p = h + . That is, we will give the following four results.
where the rst four terms are From our theorems we may immediately deduce the following: Note that the estimate α ≤ √ p, from Corollary 1.5 we also have the following: For any prime p with p ≡ mod and any positive integer k, let M k (p) denote the number of the solutions of the congruence equation Then from our theorems we can give an exact computational formula for M k (p). For example, let H s (p) denote the number of the congruence equation Then we have the identity Since H k (p) has a fourth-order linear recurrence formula (see [8]), so from the above formula and our theorems we can deduce the exact value of M k (p).

Several lemmas
To complete the proofs of our theorems, we need to prove four simple lemmas. Hereafter, we will use many properties of the classical Gauss sums and the fourth-order character mod p, all of which can be found in books concerning Elementary Number Theory or Analytic Number Theory, such as references [7], [14] or [15]. Some important results related to Gauss sums can also be found in [16] and [17]. These contents will not be repeated here. First we have the following: Lemma 2.1. Let p be a prime with p ≡ mod , λ be any fourth-order character mod p, then we have a p denotes the classical Gauss sums, and * p is the Legendre's symbol mod p.
Proof. In fact this is Lemma 2 of [18], so its proof is omitted.
Let p be a prime with p ≡ mod , then for any fourth-order character λ mod p, we have the identity where α is the same as in Lemma 2.1.
Now we calculate each term in (13). If p ≡ mod , then note that λ(− ) = − we have Applying (6) It is clear that the congruences a + b + ≡ mod p and a + b + ≡ mod p implies that ab ≡ mod p and a ≡ b ≡ mod p with a ≠ b. So we have Applying (13), (14), (15) and (16) we have the identity If p ≡ mod , then we also have Applying (13), (18), (19) and (20) we have It is clear that Lemma 2.3 follows from (17) and (21).

Proofs of the theorems
So if p = h + , then from (28), Lemmas 2.1-2.4 we have If p = h + , then from Lemmas 2.1-2.4 we have Applying (28) and the method of proving (29) we also have Similarly, if p = h + , then we have Finally, note that if p = h + , then from (6)