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Some new hybrid power mean formulae of trigonometric sums
Advances in Difference Equations volume 2020, Article number: 220 (2020)
Abstract
We apply the analytic method and the properties of the classical Gauss sums to study the computational problem of a certain hybrid power mean of the trigonometric sums and to prove several new mean value formulae for them. At the same time, we also obtain a new recurrence formula involving the Gauss sums and two-term exponential sums.
1 Introduction
For any integer m and odd prime \(p\ge 3\), the cubic Gauss sums \(A(m, p)=A(m)\) are defined as follows:
where, as usual, \(e(y) = e^{2\pi i y}\).
We found that several scholars studied the hybrid mean value problems of various trigonometric sums and obtained many interesting results. For example, Chen and Hu [1] studied the computational problem of the hybrid power mean
where c̅ denotes the multiplicative inverse of \(c\bmod p\), that is, \(c\cdot \overline{c}\equiv 1\bmod p\).
For \(p\equiv 1\bmod 3\), they proved an interesting third-order linear recurrence formula for \(S_{k}(p)\).
Li and Hu [2] studied the computational problem of the hybrid power mean
and proved an exact computational formula for (1).
Zhang and Zhang [3] proved the identity
Other related contents can also be found in [4–12], which will not be repeated here.
In this paper, inspired by [1] and [2], we consider the following mean value:
We do not know whether there exists a precise computational formula for (2), where c is any integer with \((c, p)=1\), and \(p\equiv 1\bmod 3\).
Actually, there also exists a third-order linear recurrence formula of \(H_{k}(c, p)\) for all integers \(k\geq 1\) and c. But for some integers c, the initial value of \(H_{k}(c, p)\) is very simple, whereas for other c, the initial value of \(H_{k}(c, p)\) is more complex. So a satisfactory recursive formula for \(H_{k}(c, p)\) is not available.
The main purpose of this paper is using an analytic method and the properties of classical Gauss sums to give an effective calculation method for \(H_{k}(c, p)\) with some special integers c. We will prove the following two theorems.
Theorem 1
Letpbe a prime with\(p\equiv 1\bmod 3\). If 3 is not a cubic residue\(\bmod~p\), then we have
and
Theorem 2
Letpbe an odd prime with\(p\equiv 1\bmod 3\). If 3 is a cubic residue\(\bmod~p\), then for any integer\(k\geq 3\), we have the third-order linear recurrence formula
where the first three terms are\(H_{0}(1,p)=2p^{2}-pd\), \(H_{1}(1,p)=p^{2} (d-6 )\), and\(H_{2}(1,p)=p^{2}(6p-5d)\).
Some notes: First, in Theorem 1, if \((3,p-1)=1\), then the question we are discussing is trivial, because in this case, we have
Second, in the first and third formulas of Theorem 1, we take \(c=3\) (and \(c=1\) in the second formula). These are all for getting the exact value of the mean value. Otherwise, the results will not be pretty.
2 Several lemmas
To complete the proofs of our theorems, several lemmas are essential. Hereafter, we will use related properties of the classical Gauss sums and the third-order character \(\bmod~p\), all of which can be found in books concerning elementary number theory or analytic number theory, such as [13] and [14]. First we have the following:
Lemma 1
Letpbe a prime with\(p\equiv 1\bmod 3\). Then for any third-order character\(\psi \bmod p\), we have the identity
Proof
First, applying the trigonometric identity
and noting that \(\psi ^{3}=\chi _{0}\), the principal character \(\bmod~p\), we have
Noting that \(\psi ^{2}=\overline{\psi }\) and \(\tau (\psi )\tau (\overline{\psi } )=p\), from the properties of Gauss sums we have
Since ψ is a third-order character \(\bmod~p\), for any integer c with \((c,p)=1\), from the properties of the classical Gauss sums we have
Applying (7), we have
Combining (4), (5), (6), and (8), we have the identity
This proves Lemma 1. □
Lemma 2
Letpbe a prime with\(p\equiv 1\bmod 3\), and letψbe any third-order character\(\bmod~p\). Then we have
where\(\tau (\psi )\)denotes the classical Gauss sums, anddis uniquely determined by\(4p=d^{2}+27b^{2}\)and\(d\equiv 1\bmod 3\).
Proof
Lemma 3
Letpbe a prime with\(p\equiv 1\bmod 3\). Then we have the identity
Proof
Since the congruence equation \(x^{3}+1\equiv 0\bmod p\) has three solutions in a reduced residue system \(\bmod~p\), from (3) we have
It is clear that the conditions \(a^{3}+b^{3}+1\equiv 0\bmod p\) and \(a+b+1\equiv 0\bmod p\) (\(0\leq a\), \(b\leq p-1\)) imply \(a(a+1)\equiv 0\bmod p\) and \(a+b+1\equiv 0\bmod p\), or \((a, b)=(0,p-1)\) and \((a, b)=(p-1, 0)\). So we have
From (3), (7), Lemma 2, and the properties of Gauss sums we have
Combining (9), (10), and (11), we have the identity
This proves Lemma 3. □
3 Proofs of the theorems
We achieve our main results in this part. First, we prove Theorem 1. For any integer m with \((m, p)=1\), from (7) and Lemma 2 we have
Applying (7) and Lemmas 1 and 2, we have
where we have used the identity \(1+\psi (3)+ \overline{\psi }(3)=0\).
Applying Lemmas 1, 2, and 3 and (7), we have
Applying Lemmas 1, 2, and 3 and (12), we have
Now Theorem 1 follows from (13), (14), and (15).
If \(p\equiv 1\bmod 3\) and 3 is a cubic residue \(\bmod~p\), then \(\psi (3)=\overline{\psi }(3)=1\). From Lemma 3 we have
From (7) and Lemmas 1 and 2 we have
From (7) and Lemmas 1, 2, and 3 we also have
If \(k\geq 3\), then applying (12), we have
Now Theorem 2 follows from (16), (17), (18), and (19).
This completes the proofs of all our results.
4 Conclusion
The main work of this paper includes two theorems. In Theorem 1, we obtained some exact values of (2) when \(k=1, 2\), and 3. In Theorem 2, we showed that \(H_{k}(1,p)\) satisfies an interesting third-order linear recurrence formula. These works not only profoundly reveal the regularity of a certain hybrid power mean of the trigonometric sums, but also provide some new ideas and methods for further study of such problems.
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Acknowledgements
The authors would like to thank the Editor and referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
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This work is supported by the N.S.F. (11771351 and 11826205) of P.R. China.
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Chen, L., Chen, Z. Some new hybrid power mean formulae of trigonometric sums. Adv Differ Equ 2020, 220 (2020). https://doi.org/10.1186/s13662-020-02660-7
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DOI: https://doi.org/10.1186/s13662-020-02660-7