On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

Convergence of generalized Kapteyn expansion

Series expansions of Kapteyn type in terms of Gamma function and a generalization of Bessel function of first kind were considered by Exton. Here we show that the convergence conditions given by him can be relaxed and the achieved results are optimal.     

On the sums of a family of Kapteyn series

The series ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) arise in studying the emission and absorption of radiation by a charged particle on a Kepler orbit. For the first few even,r, the sums are obtained in closed form, and for oddr they are given in terms of a certain definite integral. The integral is expressed as a power series in for ||<1, and, near =1, an asymptotic expansion in powers of (1–²)^1/2 may be obtained.

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Résumé La série ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) se trouve par l'émission et l'absorption du rayonnement d'une particule chargée sur l'orbite Keplerien. Pour les plus petites valuers paires der, les sommes s'obtienment en forme compacte, et pour les valuers impaires, elles se déterminent d'après une intégrale definie. Pour ||<1, cette intégrale se développe dans une série de puissances de , et dans le voisinage de =1, on obtient une série asymptotique et puissances de (1–²)^1/2.

     

On rational addition theorems

We investigate the functions which admit an addition theorem of the form

     

On linear versions of some addition theorems

Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dim _K ?AB? and dim _K ?B ⁿ ? in terms of dim _K A, dim _K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.     

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.     

Existence theorems for optimal control problems involving functional differential equations

This paper extends some existence theorems of Cesari for optimal control problems to systems whose dynamics is described by functional differential equations of finitely-retarded type. We show that the proper choice of state space is the spaceE ¹×C[–, 0], where >0 represents the time-lag of the system, and that it is necessary to choose initial conditions from a compact set inC[–, 0] as well as to employ the usual growth condition.This research was accomplished in the frame of research project AFOSR-942-65 at the University of Michigan. In particular, the author would like to thank Professor L. Cesari (University of Michigan) and Professor N. Chafee (Brown University) for many helpful remarks during the preparation of the research, which forms part of the author's doctoral dissertation written at the University of Michigan.     

Embedding theorems for two-parameter groups of formal power series and related problems

The classical problem of analytic iteration is that of embedding analytic functions in one-parameter Lie groups of formal power series. The main purpose of the present paper is to consider similar problems for two-parameter groups. These problems are closely related to problems concerning conjugate power series and conjugate one parameter groups of such series, to which the first sections of the paper are devoted. As an application of the conjugacy theorems and the embedding theorems we bring an algebraic characterization of the class of the two-parameter groups.     

On examples concerning integrability theorems for trigonometric series

Examples of trigonometric series are constructed that are not Fourier series, with coefficients whose differences are majorized by an arbitrary sequence of numbers given in advance, which satisfies certain natural conditions.     

On the kolmogorov theorems on Fourier series and conjugate functions

We consider Fourier series of summable functions from spaces ??wider?? than L ₁. We describe classes ??(L) which contain conjugate functions, where their conjugate Fourier series converge. The obtained results are more general than A. N. Kolmogorov theorems on the convergence of Fourier series in metrics weaker than that of L ₁.     

On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.     

Vector addition theorems and Baker-Akhiezer functions

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of anN-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rankN for a function of ag-dimensional argument that generalize the classical functional Cauchy equation. It is shown that forN=2 the general analytic solution of these equations is determined by the Baker—Akhiezer function of an algebraic curve of genus 2. It is also shown that functions give solutions of a Cauchy equation of rankN for functions of ag-dimensional argument withN2 ^g in the case of a generalg-dimensional Abelian variety andNg in the case of a Jacobian variety of an algebra curve of genusg. It is conjectured that a functional Cauchy equation of rankg for a function of ag-dimensional argument is characteristic for functions of a Jacobian variety of an algebraic curve of genusg, i.e., solves the Riemann—Schottky problem.In memory of M. K. PolivanovResearch Institute of Physicotechnical and Electronic Measurements; L. D. Landau Institute of Theoretical Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 200–212, February, 1993.     

An application of Kapteyn series to a problem from queueing theory

Using results from the theory of Kapteyn series, we solve a problem arising from the heavy traffic analysis of a Markovmodulated queue model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)