#### Abstract

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.

#### 1. Introduction

As usual, let be an odd prime. For any integer , we define the cubic Gauss sums as follows:where and .

In recent years, several scholars have studied the hybrid power mean problems of various trigonometric sums and proved many interesting results. For example, Chen and Hu  studied the computational problem ofwhere denotes the multiplicative inverse of . That is, . For , they obtained a third-order linear recurrence formula for .

Li and Hu  studied another hybrid power meanand gave an exact computational formula for (3).

Some other related papers can also be found in references . We would not have to repeat them here.

Very recently, Chen and Chen  studied the recursive properties of the hybrid power meanand obtained a third-order linear recurrence formula for it with and . That is, they proved the following result:

Let be an odd prime with . If 3 is a cubic residue , then for any integer , one has the third-order linear recurrence formula:where the first three terms are , , and .

Note that Zhang and Zhang  proved the identity

Perhaps this is the best result for which there is no conditional requirement on the prime . The interesting results of the above work motivate us to ask such a problem of whether there exists a similar recursive formula for the hybrid power meanwhere is an odd prime with .

Obviously, the problem in (7) is much harder than the problem in  because we are dealing with the fourth power mean of the two-term exponential sums in (7). Our main contribution is to obtain an identity for the fourth power mean of the two-term exponential sums weighted by third-order character modulo , i.e., the following Lemma 3. Then, we use this lemma to derive several interesting recursion formulas for . In this way, the continuities and the value distribution properties of this kind of trigonometric sums can be described from different views. Of course, the reason why we focus on the calculation of (7) is that the problem is closely related to the number of the solutions of some congruence equation. These contents play a very important role in study of some famous analytic number theory problems, such as Waring problem and Goldbach conjecture.

Through the study, it is found that the problem we studied is closely related to integer 3. If 3 is a cubic residue modulo , then there exists a beautiful third-order linear recurrence formula for , and the first three terms , , and are integers. If 3 is not a cubic residue , then we can get the exact value of . For any other positive integer , we can only give a more complex mathematical representation for . That is, we have the following three results:

Theorem 1. Let be a prime with . If 3 is a cubic residue modulo , then for any integer , we have the third-order linear recurrence formulawhere the first three terms are , , and and is uniquely determined by and .

Theorem 2. Let be a prime with . If 3 is a cubic residue modulo , then for any integer , we also have the third-order linear recurrence formulawhere the first three terms are , , and .

Theorem 3. Let be an odd prime with . If 3 is not a cubic residue modulo , then we have the identity

From our theorems, we may immediately deduce the following three corollaries:

Corollary 1. Let be an odd prime with , then we have

Corollary 2. Let be a prime with . If 3 is a cubic residue modulo , then we have

Corollary 3. Let be a prime with . If 3 is a cubic residue modulo , then we have

Some notes: first in Theorem 2, if , then the question we are discussing is trivial. Because in this case, we have

Second, if and 3 is not a cubic residue modulo , then we can only get the exact value of .

Third, the advantage of our work is that we completely solve the calculation problem of with .

Fourth, the mean value estimation of the exponential sums is closely related to the upper and lower bounds of the individual exponential sums. So, by studying the mean value of the positive exponential sums, we can obtain a better upper bound estimation of the exponential sums. If we want to get its lower bound estimation of the exponential sums, we should study the negative power of the exponential sums. Our Theorems 1 and 2 address both types of problems.

Finally, for any fixed positive integer , whether there is a third-order linear recurrence formula for the hybrid power meanis an open problem, which is the limitation of our work. The other drawback, of course, is that we cannot compute all when 3 is not a cubic residue modulo . In fact, our ultimate goal is to obtain a precise calculation formula for for all positive integers . In the future, we will continue to improve the research in this aspect. It also requires us to continue to study.

#### 2. Several Lemmas

To complete the proofs of our theorems, several simple lemmas are necessary. Hereafter, we will use many properties of the classical Gauss sums and the third-order character modulo , all of which can be found in books concerning about Elementary Number Theory or Analytic Number Theory, such as references , so the related contents will not be repeated here. First we have the following:

Lemma 1. If is a prime with , then for any third-order character , we have the identity

Proof. First applying trigonometric identityand noting that , the principal character modulo , we haveNote that and , and from the properties of Gauss sums and the characteristic function of the third-order character modulo we haveSince is a third-order character modulo , for any integer with , from the properties of the classical Gauss sums, we haveFrom (22) and the properties of Gauss sums, then we can getCombining (18), (20), (21), and (23), we have the identityThis proves Lemma 1.

Lemma 2. If p is a prime with and is any third-order character modulo , then we have the identitywhere denotes the classical Gauss sums, is uniquely determined by , and .

Proof. See  or .

Lemma 3. If is a prime with , then for any third-order character , we have the identity

Proof. Note that the two-term exponential sums satisfiesSo, from the properties of Gauss sums and Lemma 1, we haveFrom (22) and Lemma 2, we haveIt is clear that is a real number, so from the properties of Gauss sums and (29), we haveNote that the congruence implies the congruenceSo, we haveIt is clear thatFrom (32) and (33), we haveCombining (28), (30), (32), and (34), we may immediately deduce thatThis proves Lemma 3.

Lemma 4. If is a prime with , then we have the identity

Proof. See Zhang and Zhang .

#### 3. Proofs of the Theorems

Now, we prove our main results. First, we prove Theorem 1. If 3 is a cubic residue , then . From (22), Lemmas 2 and 3, we have

Applying (22), Lemmas 24, we have

Applying (29), (37), Lemmas 3 and 4, we also have

If , then from (29) and the definition of , we have

Combining Lemma 4, (37)–(40), we complete the proof of Theorem 1.

Note that identity , so we have , , and .

From Lemma 4 and Theorem 1, we have the third-order linear recurrence formula:

If , then we have the third-order linear recurrence formula

This proves Theorem 2.

Now we prove Theorem 3. If and 3 is not a cubic residue modulo , then we have or . From (22), Lemmas 24, we have

This completes the proofs of all our results.

#### 4. Conclusion

This paper mainly proposed three theorems, which are all closely related to the hybrid power mean of the two-term exponential sums and the classical cubic Gauss sums. Theorems 1 and 2 obtained two interesting third-order linear recurrence formulas of for and , respectively, providing 3 is a cubic residue modulo . However, when 3 is not a cubic residue modulo , we gave an exact calculating formula for in Theorem 3. In general, this work not only generalized the results in reference  but also provided some new ideas and methods for the further study of such problems.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Authors’ Contributions

All authors have equally contributed to this work. All authors read and approved the final manuscript.

#### Acknowledgments

This work was supported by the Y. S. T. N. S. P (2019KJXX-076), the N. S. B. R. P. (2019JM-207) of Shaanxi Province, and the N. S. F. (11771351) of P. R. China.