Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method. Received 25 August 2000; in revised form 8 January 2002     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method.     

Transcendence of Certain Trigonometric Series

In this paper the transcendence of values of certain trigonometric series with algebraic coefficients and its derivatives at algebraic points is proved under a general condition dependent only on the coefficients. The proof of the theorem is based on a criterion for linear independence over a number field. Received November 9, 1999, Accepted February 5, 2001     

某些无穷乘积的超越性

本文给出某些用无穷乘积表示的函数在代数点和超越点上的值的超越性．     

某些三角级数的超越性

本文在仅与系数有关的一般条件下证明了某些代数系数三角级数及其导数在代数点上值的超越性.定理的证明基于一个关于数域上的线性无关性的判别法则.     

Sums of Reciprocal Eigenvalues of the Laplacian

The purpose of this paper is to present formulas for the sum of the squares of all reciprocal eigenvalues λ ofthe fixed membrane and the free membrane μ for plane simply connected domains. As a consequence we obtain that among all simply connected plane domains with the maximal conformal radius 1 and the area A only the unit disk yields the minimum of .     

A Diophantine Equation Concerning Linear Recurrences

Periodica Mathematica Hungarica -     

Digital Sums and Divide-and-Conquer Recurrences: Fourier Expansions and Absolute Convergence

We propose means for computing the Fourier expansions of periodic functions appearing in higher moments of the sum-of-digits function and in the solutions of some divide-and-conquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently compute numerically the coefficients involved to high precision.     

一类加性函数的倒数和

设ｇ（ｎ）＞０为任一实值函数,加性函数Ｇ（ｎ）定义为Ｇ（１）＝０,Ｇ（ｎ）＋ｋ≥１为一固定正整数。本文研究了和式∑２≤ｎ≤ｘ　,并得到了渐近公式。     

Linear Codes and Character Sums

Let V be an rn-dimensional linear subspace of . Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces.First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction of vectors in V with weight d is exponentially small. Specifically, in the interesting case of a small r, this fraction does not exceed .We also answer a question of Ben-Or and show that if , then for every k, at most vectors of V have weight k.Our work draws on a simple connection between extremal properties of linear subspaces of and the distribution of values in short sums of -characters.* Supported in part by grants from the Israeli Academy of Sciences and the Binational Science Foundation Israel-USA. This work was done while the author was a student in the Hebrew University of Jerusalem, Israel.     

On the Transcendence of Infinite Sums of Values of Rational Functions

Convergent sums and , where P(X), Q(X) Q[X], and Q(X)has only simple rational roots, are investigated. Adhikari,Shorey and the authors have shown that T and U are either rationalor transcendental. Simple necessary and sufficient conditionsare formulated for the transcendence of T and U if the degreeof Q is 3 and 2, respectively.     

Fourth Order Linear Recurrences Satisfied by Wythoff Pairs

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs , where i is a natural number and is the golden ratio . It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if is adjoined to the rationals.     

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.     

A Positive Functional Bound for Certain Sine Sums

The positivity of the sine sums

分式线性递推数列极限的换元解法

给出由关系式xn＋1=（axn＋b）/（cxn＋d）所确定的分式线性递推数列极限的换元解法.     

Transcendence Criterion for Values of Certain Functions of Several Variables

Let ${\Phi_0(\boldmath{z})}$ be the function defined by $$\Phi_0({\boldmath z}) = \Phi _{0}(z_1,\ldots, z_m)=\sum_{k\geq 0}\frac{E_k(z_1^{r^k},\ldots,z_m^{r^k})}{F_k(z_1^{r^k},\ldots,z_m^{r^k})},$$ where ${E_k(\boldmath{z})}$ and ${F_k(\boldmath{z})}$ are polynomials in m variables ${\boldmath{z} = (z_1,\ldots, z_m)}$ with coefficients satisfying a weak growth condition and r ≥ 2 a fixed integer. For an algebraic point ${\boldmath{\alpha}}$ satisfying some conditions, we prove that ${\Phi_{0}(\boldmath{\alpha})}$ is algebraic if and only if ${\Phi_{0}(\boldmath{z})}$ is a rational function. This is a generalization of the transcendence criterion of Duverney and Nishioka in one variable case. As applications, we give some examples of transcendental numbers.     

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.     

Convergence almost Everywhere of Certain Partial Sums of Fourier Integrals

Suppose that R goes to infinity through a second-order lacunaryset. Let S_R denote the Rth spherical partial inverse Fourierintegral on R^d. Then S_Rf converges almost everywhere to f, providedthat f satisfies . 2000 Mathematics SubjectClassification 42B15, 42B25, 42B08.