Book review

Hypergeometric series identities are revisited systematically by means of Abel's method on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel's method on summation by parts is a natural choice in dealing with classical hypergeometric series.     

Summation formulae for twisted cubic q‐series

A new class of twisted cubic q‐series is investigated by means of the modified Abel lemma on summation by parts. Several remarkable summation and transformation formulae are established for both terminating and nonterminating series.     

Abel’s lemma on summation by parts and partial <Emphasis Type="Italic">q</Emphasis>-series transformations

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established. This work was partially supported by National Natural Science Foundation for the Youth (Grant No. 10801026)     

An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications

A simple proof is given of a new summation formula recently added in the literature for a terminating _{r + 3}F_{r + 2}(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well‐known Saalschütz summation formula for a ₃F₂(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the ₂F₁ series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijovi? & Miller. Copyright © 2015 John Wiley & Sons, Ltd.     

A method of summation of divergent series to any accuracy

We propose a new method of summation to any accuracy for a wide class of divergent series, using only a finite number of terms of the series. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 24–35, July, 2000.     

Applications of hypergeometric summation theorems of kummer and dixon involving double series

Using series iteration techniques, we derive a number of general double series identities and apply each of these identities in order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.     

The disteribution of a moment estimator for a parameter of the generalized poision distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Reduction Formulas for Karlsson–Minton-Type Hypergeometric Functions

We prove a master theorem for hypergeometric functions of Karlsson–Minton type, stating that a very general multilateral U(n) Karlsson–Minton-type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson–Minton summation formula and many of its known generalizations as special cases, and it also implies several Bailey-type identities for U(n) hypergeometric series, including multivariable ₁₀W₉ transformations of Denis and Gustafson and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of multivariable hypergeometric series.     

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.     

An Euler product transform applied to q-series

This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation. 2000 Mathematics Subject Classification Primary—11M41; Secondary—33D15, 30B50     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.     

Approximation processes for fourier-jacobi expansions †

This paper deals with the approximation theoretic aspects of summation methods for expansions in terms of Jacobi polynomials. When a funcation f is expanded in a Fourier-Jacobi series, many summation methods for this series may be looked upon as approximation processes for the function f. The main object of this paper is to investigate the order of approximation of these processes and to characterize the functions which allow a certain order of approximation. Many of these processes exhibit the phenomenon of saturation, which is equivalent to the existence of an optimal order of approximation (the saturation, which is equivalent to the existence of an optimal order of approximation (the saturation order). For the approximation processes treated in this paper the saturation order and the saturation class, that is the class if functions which can be approximated with the optimal order, are derived. The characterization of the classes of functions is accomplished by means of the theory of intermediate spaces due to Peetre[19] (compare Butzer and Berens [7]). Another basic tool in this work is the convolution structure for Jacobi series, introduced by Askey and Wainger [1] (see also Gasper [14], {15})     

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189-371] or the technique recently introduced by ?í?ek et al. [J. ?í?ek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962-968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.     

A Multidimensional Generalization of Shukla's 8ψ8 Summation

   Abstract. We give an r -dimensional generalization of H. S. Shukla's very-well-poised ₈ ψ ₈ summation formula. We work in the setting of multiple basic hypergeometric series very-well-poised over the root system A _r-1 or, equivalently, the unitary group U(r) . Our proof, which is already new in the one-dimensional case, utilizes an A _r-1 nonterminating very-well-poised ₆ φ ₅ summation by S. C. Milne, a partial fraction decomposition, and analytic continuation.     

Analogues of Euler and Poisson summation formulae

Euler-Maclaurin and Poisson analogues of the summations ε _{a <n ≤b} χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula in several variables.     

Theorems on unions ofU-sets

The properties of families of subsets of topological spaces that make it possible to separate out the common core of the proofs of theorems on unions of sets of uniqueness for series in various systems of functions are considered. As a consequence of the results obtained, it is proved, in particular, that the countable union of closed sets of uniqueness for the trigonometric series is a set of uniqueness for the Abel-Poisson summation method. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 778–787, May, 2000.     

Linear Series on 4 – Gonal Curves

Let C be the general 4 – gonal curve of genus g > 6. We investigate the complete linear series on C and the varieties W^r_d(C), and we study the birational models of C in ℙ^r (r ≥ 2) of minimal degree.     

双变量基本超几何级数的若干变换与求和公式

The purpose of this paper is to establish several transformation formulae for bivariate basic hypergeometric series by means of series rearrangement technique. From these transformations, some interesting summation formulae are obtained.     

Heegner Divisors and Nonholomorphic Modular Forms

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.     

Some Closed-Form Evaluations of Multiple Hypergeometric and <Emphasis Type="Italic">q</Emphasis>-Hypergeometric Series

The main object of the present paper is to show how some fairly general analytical tools and techniques can be applied with a view to deriving summation, transformation and reduction formulas for multiple hypergeometric and multiple basic (or q-) hypergeometric series. By making use of some reduction formulas for multivariable hypergeometric functions, the authors investigate several closed-form evaluations of various families of multiple hypergeometric and q-hypergeometric series. Relevant connections of the results presented in this paper with those obtained in earlier works are also considered. A number of multiple q-series identities, which are developed in this paper, are observed to be potentially useful in the related problems involving closed-form evaluations of multivariable q-hypergeometric functions. Dedicated to the Memory of Leonard Carlitz (1907–1999)Mathematics Subject Classifications (2000) Primary 33C65, 33C70, 33D70; secondary 33C20, 33D15.