Zeroes of slice monogenic functions

In this article, we study the structure of zeroes of power series with Clifford algebra‐valued coefficients. Especially, if it has paravector‐valued coefficients, we obtain some sufficient and necessary conditions of power series that have zeroes, as well as a method to compute the zeroes if exist. Copyright 2011 John Wiley & Sons, Ltd.     

Arithmetic Continuation of Regular Roots of Formal Parametric Polynomial Systems

Given a regular system of polynomial equations with power series coefficients, an initial root is continued as a power series. With the ground domain as an arbitrary field, arithmetic alone is used for the root continuation over this field, and computation is quadratic in the number of computed coefficients. If the power series of the coefficients of the polynomial are geometrically bounded, then the coefficients of the power series of the root are also.     

Density problems connected with overconvergence

Due to Ostrowskii’s classical results on overconvergence of power series, there is an interdependence between the occurrence of gaps in the sequence of coefficients and the overconvergence phenomenon. The paper investigates the structure of the corresponding gaps and especially the density properties of gap intervals, as well as of “non-gap” intervals of overconvergent power series.     

Small Perturbation of Stochastic Parabolic Equations: A Power Series Analysis

A semi-linear second-order stochastic parabolic equation is considered with coefficients, free terms, and initial condition depending on a parameter. It is shown that under some natural conditions the solution can be written as a power series in the parameter. The equations for the coefficients in the power series expansion are derived and the convergence of the power series is studied. An example from nonlinear filtering of diffusion process is discussed.     

关于两类幂级数系数的重排

该文研究了两类幂级数的系数重排后的增长性，得到了平面上和单位圆内有限级幂级数的系数经过重排后其级和型不变的一些充要条件．     

Applications of?the Cosecant and Related Numbers

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c _k , converge rapidly to zero as k????. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc?(at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d _k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.     

关于Boussinesq方程Barenblatt幂级数解的注记

通过幂级数展开的方法推求得出了Barenblatt幂级数解的各项系数之间的递推公式(对半无限长多孔介质中地下水流动的Boussinesq方程的自相似解,在边界水头随时间幂函数变化的条件下,Barenblatt（1952）得到了一个幂级数解,但他仅仅列出了其前3项的系数,既没有给出整个幂级数解所有系数的递推关系式,也没有证明该幂级数解的收敛性．),并对该级数的收敛性进行了证明,同时对解的实际应用作了讨论．这些研究结论易于理解,方便工程技术人员应用于流域水文学和基流研究及解决农业排水等实际问题．     

Factorization of matrix functions and their inverses via power product expansions

The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinite product of elementary factors is also analyzed. Each elementary factor is the sum of the identity matrix and a certain matrix coefficient multiplied by a certain power of the variable. The two expansions provide us with representations of a matrix function and its inverse by infinite products of elementary factors. Estimates on the domain of convergence of the infinite products are given.     

Reversion of power series and the extended Raney coefficients

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

     

A poset structure on quasifibonacci partitions

In this paper, we study partitions of positive integers into distinct quasifibonacci numbers. A digraph and poset structure is constructed on the set of such partitions. Furthermore, we discuss the symmetric and recursive relations between these posets. Finally, we prove a strong generalization of Robbins' result on the coefficients of a quasifibonacci power series.     

幂级数系数重排使型不变的必要条件

根据幂级数系数重排级不变的充要条件，对比研究了幂级数系数的重排与此级数的和函数的型之间的关系，得到了幂级数系数重排型不变的一些必要条件。     

Die exakte Lösung der Integralgleichungen gewisser Schwingungsprobleme

Summary The vibration problem of certain dynamic systems with polynomial mass and stiffness distributions can be expressed as a Fredholm integral equation with a degenerated, symmetric kernel.If a starting function is chosen appropriately, the eigenfunction can be expressed as a power series. Simple recurrence relations between the coefficients of this power series yield the characteristic equation for the eigenvalues with a finite number of disposable coefficients.This method is applied to a beam and a wedge and leads to the exact solutions.     

Convergent expansions in non-relativistic qed: Analyticity of the ground state

We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schrödinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant α, with α dependent coefficients which are finite for α?0.     

BMO- andL p -conditions for power series and Dirichlet series with positive coefficients

It is known that theL _p -norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients are nonnegative. In the paper we prove analogous estimates for theL _p -norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums of power series and Dirichlet series, respectively, by means of their coefficients.     

Asymptotic Analysis of Solutions to a Riccati Equation

A Riccati equation with coefficients expandable into convergent power series in a neighborhood of infinity is considered. Continuable solutions to equations of this type are studied. Conditions for the expansion of these solutions into convergent series in a neighborhood of infinity are obtained by methods of power geometry.     

The nil radical of power series rings

We describe the nil radical of power series rings in non-commuting indeterminates by showing that a series belongs to the radical if and only if the ideal generated by its coefficients is nilpotent. We also show thatt the principal ideals generated by elements of the nil radical of the power series ring in one indeterminate are nil of bounded index.     

斯泽古定理的历史研究

探讨幂级数在收敛圆上的行为表现是函数解析开拓的一个重要问题,"具有有限多个不同系数的幂级数"是其研究的重要一类,斯泽古定理即是该类级数研究的一个重要成果.文章基于原始文献,利用历史分析和比较的方法,探讨了斯泽古定理提出的思想背景,法都猜想是其重要的思想来源,详细分析了该定理的形成过程及进一步的发展,对深入理解斯泽古定理的发展历史具有重要作用.     

Computation of the Chebyshev-Padé table

A recursive method is given for the computation of the coefficients in the Chebyshev-Padé table. This is a table, recently defined by Clenshaw and Lord for Chebyshev series, which is analogous to the Padé table for power series. The method enables one to compute the whole of the triangular part of the table which derives from the given number of terms in the original Chebyshev series taken into account. The recursive method given by Clenshaw and Lord only enables one to compute the coefficients in half of this table.     

Irregular points of modulation

The local theory of singular points is extended to a large class of linear, second-order, ordinary differential equations which can be physical Schroedinger equations, or govern the modulation of real oscillators or waves. In addition to Langer's fractional turning points, such equations admit highly irregular points at which the coefficients of the differential equation can be almost arbitrarily multivalued. (Genuine coalescence of singular points, however, is not considered.) A local representation of the solutions is established, which generalizes Frobenius' method of power series, and reveals a remarkable, two-variable structure. Bounds are obtained on the departure of solution structure from the structure characteristic of regular points.     

Automation of the manipulation of multivariate power series

This paper reports on the development of compact and remarkably general algorithms for the manipulation of multivariate power series. The problem of efficiently storing the coefficients of such series is solved in a way which admits weighted truncation and yields simple algorithms for (i) algebraic operations, (ii) composition of special functions with power series and (iii) composition and reversion of multivariate power series. The algorithms, which are expressed in a form that can readily be translated into any standard computer language, can manipulate power series in an arbitrary number of variables while retaining all terms up to an arbitrary weighted order with respect to an arbitrary set of weights. The size of the power series which can be manipulated is limited only by memory capacity. For most purposes, a conventional microcomputer is adequate.