形式幂级数软件包

关于正交多项式和特殊函数的软件包

A package on orthogonal polynomials and special functions

本文介绍 Mathematica 软件包 SpecialFunction，其核心功能是幂级数与生成函数转换、自动求微分和递推方程，并支持高效计算经典正交多项式，用于快速求解线性微分方程。

On the convergence of an exotic formal series solution of an ODE

A sufficient condition of the convergence of an exotic formal series (a kind of power series with complex exponents) solution to an ODE of a general form is proposed.

On the convergence of formal Dulac series satisfying an algebraic ODE

We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painlevé equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.

On the convergence of formal power-log series solutions of an algebraic ODE

We propose a sufficient condition of the convergence of a power-log series that formally satisfies an algebraic ordinary differential equation (ODE) of arbitrary order. A general form and properties of the functional coefficients of such a series are established.

形式幂级数

Formal power series

在这篇文章中，我们将描述的\Maple-\cite{Koeortho}中提出的一个算法的\Maple \实现，它计算一个给定的函数的{ \em精确\/}形式幂级数（FPS）。该程序将使用户能够再现系列~ \cite{Han}的大部分广泛参考书目的结果。我们将概述该算法，然后更详细地介绍其中的某些部分。

A note on the convergence of multivariate formal power series solutions of meromorphic Pfaffian systems

Here we present some compliments to theorems of Gerard and Sibuya, on the convergence of multivariate formal power series solutions of nonlinear meromorphic Pfaffian systems. Their the most known results concern completely integrable systems with nondegenerate linear parts, whereas we consider some cases of non-integrability and degeneracy.