Some Identities Involving Certain Hardy Sums and Ramanujan Sum

The main purpose of this paper is, using the mean-value theorem of Dirichlet 1-functions, to study the distribution properties of the hybrid mean value involving certain Hardy sums and Ramanujan sum, and give four interesting identities.     

Gauss和与广义Kloosterman和的几个新的恒等式

主要应用Gauss和的性质和解析方法来研究Gauss和与广义Kloosterman和之间的关系,并且给出了几个有趣的恒等式.     

三次高斯和与Kloosterman和的线性递推公式

应用三角和方法以及高斯和的若干性质,研究三次高斯和与Kloosterman和的一类高次混合均值的计算问题,本文给出该混合均值的一个有趣的线性递推公式.同时,还应用该递推公式,得到三次高斯和与Kloosterman和的高次混合均值的一系列较强的渐近公式.     

On Nonlinear Kloosterman Sums

Doklady Mathematics - Additive shift is a widely used tool for estimating exponential sums and character sums. According to it, the summation variable n is replaced by an expression of the type n +...     

On Twisted Kloosterman Sums

For the Kloosterman sums twisted by characters over a finite field, addition formulas of convolution type are derived. As a corollary, orthogonality relations connecting the Kloosterman and Salie vectors are obtained. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 96–106.     

关于Hardy和的混合均值公式

本文的目的是利用Dirichlet L-函数的均值定理研究类似于Dedekind和的和与Ramanujao和的混合均值的分布性质，并给出一个较为精确的渐近公式及一个有趣的恒等式．     

On the General Kloosterman Sums

The paper presents a summation formula for the general Kloosterman sums, which generalizes the known Kuznetsov formula. Bibliography: 22 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 107–134.     

若干包含Laguerre多项式的等式和同余式

令L_n(x)为Laguerre多项式,即L_0(x)=1,L_1(x)=-x+1,且对所有整数n≥1,有递推公式L_(n+1)(x)=(2n+1-x)L_n(x)-n~2L_(n-1)(x).主要使用组合及初等方法研究一类包含L_n(x)的卷积和式,给出其有趣的计算公式,并得到一些包含Laguerre多项式的等式和同余式,这些结果均有着重要的应用.     

Tiling Proofs of Recent Sum Identities Involving Pell Numbers

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities. Received July 20, 2006     

On Z4-Linear Goethals Codes and Kloosterman Sums

Studying the coset weight distributions of the Z₄-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.     

Hardy和Littlewood的一个三角和

本文对Hardy和Littlewood考虑的一个有限三角和做了进一步地研究.通过充分运用Chebyshev多项式和M?bius函数的性质,建立了该有限三角和的一个有趣的恒等式,并得到了一个精确的渐近公式.     

Hardy和Littlewood的一个三角和

本文对Hardy和Littlewood考虑的一个有限三角和做了进一步地研究.通过充分运用Chebyshev多项式和Möbius函数的性质,建立了该有限三角和的一个有趣的恒等式,并得到了一个精确的渐近公式.     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

广义Dedekind和与Hardy和

本文给出了广义Dedekind和与Hardy和的定义,研究了广义Dedekind和的算术性质,并把Hardy和表示成广义Dedekind和的形式．提出了广义Subrahmanyam等式和Knopp定理,并给出了证明．     

广义二次Gauss和与广义Kloosterman和的混合均值

利用分析的方法及经典二次Gauss和的性质,研究了一类广义二次Gauss和与广义Kloosterman和的混合均值的计算问题,并得到了一个精确的计算公式.     

On a Problem of D. H. Lehmer and General Kloosterman Sums

Abstract Let q ≥ 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 ≤ b < q and (b, q) = 1, it is clear that there exists one and only one c with 0 < c < q such that bc ≡ a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc ≡ a (mod q) for 1 ≤ b, c < q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums. This work is supported by the NSF and the PSF of P. R. China     

关于两类广义多项式的一些恒等式(英文)

By means of generating function and partial derivative methods, we investigate and establish several general summation formulas involving two classes of polynomials. The general results would apply to yield some identities for the Pell polynomials and Pell-Lucas polynomials, and other general polynomials can also be recovered in this paper.     

一些包含勒让德多项式的积分恒等式

设P_n(x)表示勒让德多项式.即就是P_0(x)=1,P_1(x)=x,当n≥2时有递推关系,P_(n+1)(x)=(2n+1)/(n+1)·xP_n(x)-n/(n+1)·P_(n-1)(x).主要目的是运用初等方法以及幂级数的性质讨论一类包含P_(n)(x)的卷积的定积分计算问题,并给出一些确切的计算公式.     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

On a Finite Sum Involving Inverse Powers of Cosines

A finite sum S _m,v involving inverse powers of cosines has been studied previously by Fisher, who was able to solve the v=1 and v=2 cases exactly and provide the first term of an “asymptotic solution”. The series is re-visited here by using a completely different approach from Fisher’s generating function method. Higher order terms in decreasing powers of m ² are evaluated in the large m limit. In addition, the exact calculations for the first three integer values of v are presented. An empirical method is then devised, which yields the exact formulae for all the coefficients in S _m,v when v is an integer. Consequently, the first ten values of S _m,v are tabulated.