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。     

关于两类狄里克莱级数系数的重排

研究了两类狄里克莱级数的系数重排后的增长性,得到了全平面和半平面上有限级狄里克莱级数的系数经过重排后级和型保持不变的充要条件.     

系数的模为两两NQD序列的随机幂级数的增长性

本文研究了系数的模为两两NQD序列的B-值随机幂级数的增长性.利用两两NQD列推广的Borel-Cantelli引理及其它极限定理，在给定条件下得出其增长级和非随机幂级数的增长级有类似的性质.     

一个条件收敛级数重排项后的和

研究了以自然数倒数所构成的一个典型交错级数重排项后所得级数的收敛性及求和问题．证明了当其正项和负项均按由小到大的顺序排列后,每出现r个正项后面接t个负项的排列所得到的级数收敛,并利用幂级数求得了重排项后级数的和．     

Dirichlet级数系数的重排

本文研究了Dirichlet级数系数的重排与此级数的收敛横坐标的关系.利用Knopp-Kojima的方法,获得了在Knopp-Kojima公式下绝对收敛横坐标保持不变的重排特征.     

例谈函数幂级数展开方法与系数相等的证明

函数展开为幂级数的过程中由于采用的方法不同有时会出现不同的系数表达形式,这与幂级数展开式的唯一性并不矛盾．本文通过几例揭示了幂级数展开式中系数表达形式不同的现象．分析了幂级数展开方法之间的差异．介绍了证明系数相等的一些方法．     

B-值双随机Dirichlet级数

该文较系统地研究了一类随机系数不要求同分布的B-值双随机Dirichlet级数的收敛性与增长性.证明了这类B-值双随机Dirichlet级数与某Dirichlet级数a.s.有相同的收敛、一致收敛和绝对收敛的横坐标;并证明了这类B-值双随机Dirichlet级数与某Dirichlet级数a.s.有相同的增长级和(p,q)(R)-级,以及系数经过重排后增长级a.s.保持不变的充要条件.     

幂级数系数重排的一个性质

本文探讨了级数系数重排的一些性质，给出了｜z｜〈1内级与型不变的一个充要条件。     

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.     

广义幂级数环的本质理想和非奇异性

本文研究了广义幂级数环与其系数环在本质理想和非奇异性上的关系.利用本质理想的定义和性质,得到了广义幂级数环的左理想为本质左理想的菪干充分必要条件.在此基础上,给出了广义幂级数环为左非奇异环的充分必要条件.     

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.     

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.

     

Solution of a second-order linear differential equation with polynomial coefficients and Fuchsian point at zero

We specify the structure of the power series determining a solution of a Fuchsian second-order differential equation with polynomial coefficients in a neighborhood of zero. The power series is represented via hypergeometric functions of fractional order. The structure of the coefficients of the series is clarified.     

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.     

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.     

On the series of absolute values of blocks of trigonometric series

Conditions of integrability with a power weight are obtained for the sums of series of absolute values of blocks of sine series and cosine series under certain special conditions imposed on the coefficients of these series.     

More congruences for the coefficients of quotients of Eisenstein series

Berndt and Yee (Acta Arith. 104 (2002) 297) recently proved congruences for the coefficients of certain quotients of Eisenstein series. In each case, they showed that an arithmetic progression of coefficients is identically zero modulo a small power of 3 or 7. The present paper extends these results by proving that there are infinite classes of odd primes for which the set of coefficients that are zero modulo an arbitrary prime power is a set of arithmetic density one. A new family of explicit congruences modulo arbitrary powers of 2 is also found.     

On the Expansion of Ramanujan's Continued Fraction

If Ramanujan's continued fraction (or its reciprocal) is expanded as a power series, the sign of the coefficients is (eventually) periodic with period 5. We give combinatorial interpretations for the coefficients from which the result is immediate. We make use of the quintuple product identity, which we prove.     

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.