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

2017 ◽  
Vol 15 (1) ◽  
pp. 1251-1255 ◽  
Author(s):  
Chen Zhuoyu ◽  
Zhang Wenpeng
Keyword(s):  
Positive Integer ◽  
Fourth Order ◽  
Recurrence Formula ◽  
Character Sums ◽  
Gauss Sums ◽  
Linear Recurrence ◽  
Analytic Method ◽  
Order Linear ◽  
Computational Problem

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.

scholarly journals A Four-Order Linear Recurrence Formula Involving the Quartic Gauss Sums and One Kind Two-Term Exponential Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lan Qi ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Recurrence Formula ◽  
Gauss Sums ◽  
Linear Recurrence ◽  
Analytic Method ◽  
Power Mean ◽  
Order Linear ◽  
Hybrid Power ◽  
Hybrid Power Mean ◽  
Mathematica Software

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind hybrid power mean involving the quartic Gauss sums and two-term exponential sums and give an interesting four-order linear recurrence formula for it. As an application, we can obtain all values of this kind hybrid power mean with mathematica software.

scholarly journals On the Hybrid Power Mean of Two-Term Exponential Sums and Cubic Gauss Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Shaofan Cao ◽  
Tingting Wang
Keyword(s):  
Exponential Sums ◽  
Recurrence Formula ◽  
Gauss Sums ◽  
Linear Recurrence ◽  
Power Mean ◽  
Third Order ◽  
Trigonometric Sums ◽  
Order Linear ◽  
Hybrid Power ◽  
Hybrid Power Mean

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.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

scholarly journals On Classical Gauss Sums and Some of Their Properties

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 625 ◽  
Author(s):  
Li Chen
Keyword(s):  
Recursive Formula ◽  
Character Sums ◽  
Gauss Sums ◽  
Computational Problem ◽  
Rational Polynomials

The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.

scholarly journals A hybrid mean value of the inversion of L-functions and general quadratic Gauss sums

2002 ◽  
Vol 167 ◽  
pp. 1-15 ◽  
Author(s):  
Wenpeng Zhang ◽  
Yuping Deng
Keyword(s):  
Character Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Analytic Method ◽  
Power Mean ◽  
Hybrid Mean Value

AbstractThe main purpose of this paper is, using the estimates for character sums and the analytic method, to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and give two interesting asymptotic formulas.

scholarly journals Some New Identities of Second Order Linear Recurrence Sequences

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1496 ◽  
Author(s):  
Yanyan Liu ◽  
Xingxing Lv
Keyword(s):  
Power Series ◽  
Second Order ◽  
Linear Recurrence ◽  
Combinatorial Method ◽  
Order Linear ◽  
Computational Problem ◽  
Characteristic Roots ◽  
Recursive Sequence ◽  
Recurrence Sequences

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

scholarly journals The hybrid power mean of some special character sums of polynomials and two-term exponential sums modulo $ p $

2021 ◽  
Vol 6 (10) ◽  
pp. 10989-11004
Author(s):  
Wenpeng Zhang ◽  
◽  
Jiafan Zhang ◽  
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Analytic Method ◽  
Power Mean ◽  
Hybrid Power ◽  
Special Character ◽  
Modulo P ◽  
Hybrid Power Mean

<abstract><p>We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo $ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.</p></abstract>

The positivity problem for fourth order linear recurrence sequences is decidable

2012 ◽  
Vol 128 (1) ◽  
pp. 133-142 ◽  
Author(s):  
Pinthira Tangsupphathawat ◽  
Narong Punnim ◽  
Vichian Laohakosol
Keyword(s):  
Fourth Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Recurrence Sequences

scholarly journals On the two-term exponential sums and character sums of polynomials

2019 ◽  
Vol 17 (1) ◽  
pp. 1239-1248
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

Abstract The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.

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