Rational summation ofp-adic series

The problem of rational summation for a wide class ofp-adic convergent series is considered. Here, rational summation refers to the method of obtaining the rational sum of a power series for a rational value of its variable. A formula suitable for this summation is derived. Conditions for rational summability are obtained. Rational summation is possible only for special forms of the series. It is shown that the inverse problem of rational summation is always solvable. This is illustrated by some characteristic examples. Possible rational (adelic) summation of divergent perturbative expansions in string theory, and quantum field theory, is discussed.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Toereticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 342–353, September, 1994.     

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.     

不可数求和型Hahn-Schur定理

The classical countable summation type Hahn-Schur theorem is a famous result in summation theory and measure theory. An interesting problem is whether the theorem can be generalized to non-countable summation case? In this paper, we show that the answer is true.     

付氏变换在三角级数求和中的应用

本文建立了用付氏变换在三角级数求和中的新的重要定理,并用付氏变换的已知结果,解决了不少困难和复杂的三角级数求和问题.这是三角级数求和的新方法,作者曾用以编著了数以万计的三角级数之和的大表.许多结果都是新的.     

F.H.Jackson超几何级数公式_8φ_7的推广

将 C.Krattenthaler的矩阵反演恰当地用于初文昌的恒等式得到了 F.H.Jackson的超几何级数公式 87的推广 .     

幂级数求和函数的方法及应用

幂级数求和函数是无穷级数问题中的重点和难点,该文针对幂级数求和函数总结出其常见类型和解法,求和函数时需要注意的几个问题,以及幂级数求和函数在级数求和、求极限等方面的应用.     

On a new circular summation of theta functions

In this paper, we prove a new formula for circular summation of theta functions, which greatly extends Ramanujan's circular summation of theta functions and a very recent result of Zeng. Some applications of this circular summation formula are given. Also, an imaginary transformation for multiple theta functions is derived.     

Summation of Unordered Arrays

An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.     

Infinite summation formulas related to Riemann Zeta function from hypergeometric series

Applying Gauss and Watson’s famous hypergeometric summation theorems, the authors establish two pattern infinite summation formulas involving generalized harmonic numbers related to Riemann Zeta function.     

Frobenius Partitions and the Combinatorics of Ramanujan's 1ψ1 Summation

We examine the combinatorial significance of Ramanujan's famous summation. In particular, we prove bijectively a partition theoretic identity which implies the summation formula.     

一些超几何级数恒等式的新的证明

The Abel＇s lemma on summation by parts is employed to evaluate terminating hypergeometric series. Several summation formulae are reviewed and some new identities are established.     

Accelerating Indefinite Summation: Simple Classes of Summands

We present the history of indefinite summation starting with classics (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan) followed by modern classics (Abramov, Gosper, Karr) to the current implementation in computer algebra system Maple. Along with historical presentation we describe several “acceleration techniques” of algorithms for indefinite summation which offer not only theoretical but also practical improvement in running time. Implementations of these algorithms in Maple are compared to standard Maple summation tools.     

A new summation method for power series with rational coefficients

We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented.

     

The Ewald sums for singly,doubly and triply periodic electrostatic systems

When evaluating the electrostatic potential, periodic boundary conditions in one, two or three of the spatial dimensions are often required for different applications. The triply periodic Ewald summation formula is classical, and Ewald summation formulas for the other two cases have also been derived. In this paper, derivations of the Ewald sums in the doubly and singly periodic cases are presented in a uniform framework based on Fourier analysis, which also yields a natural starting point for FFT-based fast summation methods.     

A new hypergeometric transformation of the Rathie–Rakha type

A general transformation involving generalized hypergeometric functions has been recently found by Rathie and Rakha using simple arguments and exploiting Gauss’s summation theorem. In this sequel to the work of Rathie and Rakha, a new hypergeometric transformation formula is derived by their method and by appealing to Gauss’s second summation theorem. In addition, it is shown that the method fails to give similar hypergeometric transformations in the cases of the classical summation theorems of Kummer, Bailey, Watson and Dixon.     

Strange evaluations of Fox–Wright function

By means of inversion techniques and four known hypergeometric series identities, eight summation formulas for the Fox–Wright function are established. They can give numerous summation formulas for 2-balanced hypergeometric series when the parameters are specified.     

了解积分——求和

求一个函数的黎曼积分,实际上就是一个分割、近似代替、求和、取极限的过程.求和运算是整个积分计算的轴心.就积分四部曲中的求和问题,做一个一般性的讨论.文中使用的是分析和讨论的语言,不去追求数学语言本身的严格性.目的不仅是探讨求和这个步骤,在黎曼积分意义下具体实现的过程和隐含的内容,而且对一般的积分中的求和实现的可能性、应该满足的条件、实现的过程,以及应该注意那些基本问题,也做一点儿逻辑上的探讨.已达到以知识为媒介,提高认知能力的目的.     

On Finite Forms of Certain Bilateral Basic Hypergeometric Series and Their Applications

In this paper we derive finite forms of the summation formulas for bilateral basic hypergeometric series 3ψ3,4ψ4 and 5ψ5.We therefrom obtain the summation formulae obtained recently by Wenchang CHU and Xiaoxia WANG.As applications of these summation formulae,we deduce the well-known Jacobi's two and four square theorems,a formula for the number of representations of an integer n as sum of four triangular numbers and some theta function identities.     

关于二元三角插值多项式的线性求和问题

本文构造了一个新的求和因子,使得带有该求和因子的二元三角插值多项式对任意的被插值的二元连续周期函数f(x,y)∈C(Ω)都能在全平面上一致收敛,且达到最佳收敛阶．     

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

We derive summation formulas for a specific kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A_r extensions of Ramanujan's bilateral ₁ψ₁ sum, C_r extensions of Bailey's very-well-poised ₆ψ₆ summation, and a C_r extension of Jackson's very-well-poised ₈φ₇ summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_r, B_r, C_r, and D_r, respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper–Karlsson–Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.