On certain character sums over p ‐adic rings and their L ‐functions

In this paper we present a new method for evaluating exponential sums associated to a restricted power series in one variable modulo p^l , a power of a prime. We show that for sufficiently large l, these sums can be expressed in terms of Gauss sums. Moreover, we study the associated L ‐functions; we show that they are rational, then we determine their degrees and the weights as Weil numbers of their reciprocal roots and poles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind—Rademacher Sums

Abstract. We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind—Rademacher sums , which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind—Rademacher sums, due to Rademacher.     

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.     

Series expansions for computing Bessel functions of variable order on bounded intervals

Based on a continuity property of the Hadamard product of power series we derive results concerning the rate of convergence of the partial sums of certain polynomial series expansions for Bessel functions. Since these partial sums are easily computable by recursion and since cancellation problems are considerably reduced compared to the corresponding Taylor sections, the expansions may be attractive for numerical purposes. A similar method yields results on series expansions for confluent hypergeometric functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized explicit inversive generators of small p-weight degree

Using rational functions to generate pseudorandom number sequences is a popular research topic. In this paper, we study bounds on additive character sums of a new explicit generator based on rational functions with small p-weight degree. This extends the class of functions where a nontrivial character sum bound is known.     

Summation of multiple trigonometric series by generalized methods formulated using δ -like finite functions

We consider the generalized sums of multiple trigonometric series. We investigate the sufficient conditions of convergence of the series obtained by termwise differentiation of the series for Lebesgue integrable functions as well as the errors of approximation of functions by sequences of generalized partial sums of series.     

两个新的mock theta双重和

最近, 张和李使用一个Bailey对, 获得了五个与$q$-超几何双重和相关的mock theta 函数. 本文使用此Bailey对, 我们进一步地建立了两个新的与Appell-Lerch和及theta级数相关的mock theta双重和. 进而也获得了其中一个新的mock theta函数和经典mock theta函数之间的一些关系式.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

Evaluation of power sums of roots for systems of non-algebraic equations in ℂ n

The goal of this paper is the evaluation of power sums of roots for systems of nonalgebraic equations consisting of entire or meromorphic functions of finite orders of growth. These power sums are connected with some residue integrals different from the Grothendieck residues. As an application we evaluate sums of certain multiple series.     

Extension of Edgeworth type expansion of the distribution of the sums of I.I.D. random variables in non-regular cases

The asymptotic expansions of the distributions of the sums of independent identically distributed random variables are given by Edgeworth type expansions when moments do not necessarily exist, but when the density can be approximated by rational functions. Supported in part by the Sakkokai Foundation.     

A lower bound estimate for complete sums of characters of polynomials and rational functions

In this paper we shall study the complete Dirichlet character sums involved with some polynomials and rational functions whichare useful to the Waring's problem.     

Approximation of functions by universal sums of series in subsystems of hermite polynomials

The uniform approximation of functions from a certain class on compact sets was performed first by means of polynomials, then by partial sums of universal functional series, and further by special sums associated with functional series. In this paper we construct universal sums associated with Hermite polynomials (any function from this class on compact sets is approximated by these sums). We apply the matrix transformation method that was used earlier for constructing special sums for other functional series.     

Computing elliptic integrals by duplication

Summary Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.     

Some families of combinatorial and other series identities and their applications

The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function _pΨ_q and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered.     

Green’s Function of Ordinary Differential Operators and an Integral Representation of Sums of Certain Power Series

The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.     

H_1(D)空间的Bessel级数

本文给出关于 Ｈ１（Ｄ）空间中函数的 Ｂｅｓｓｅｌ级数的部分和用幂级数的部分和表示的一个恒等式．基于它,可以得到Ｂｅｓｓｅｌ级数部分和偏差的诸多精确估计．     

H1（D）空间的Bessel级数

本文给出关于H1（D）空间中函数的Bessel级数的部分和用幂级的部分和表示的一个恒等式，基于它，可以得到Bessel 级数部分和偏差的诸多精确估计。     

Estimates of mixed norms of functions representable by series over products of cosines and sines with multiple-monotone coefficients

Lower and upper bounds for norms of mixed functions being sums of series in products of cosine and sine functions are proved in the paper.     

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.     

Decomposable polymatroids and connections with graph coloring

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.