解析$q$ -差分方程

用二阶方程求解三阶线性差分方程

Solving Third Order Linear Difference Equations in Terms of Second Order Equations

本文提出差分算子绝对因式分解的两种算法，并给出用二阶方程求解三阶差分方程的算法，应用于OEIS序列。

差分Riccati方程解的微分超越性及Tietze处理

Differential transcendence of solutions of difference Riccati equations and Tietze's treatment

这是H. Tietze出版于1905年关于微分超越的解决方案的差异Riccati方程。本文澄清了Tietze处理的本质，使之成为纯代数的。作为应用，研究了$q$ -Airy方程.

Dimension Polynomials and the Einstein's Strength of Some Systems of Quasi-linear Algebraic Difference Equations

In this paper we present a method of characteristic sets for inversive difference polynomials and apply it to the analysis of systems of quasi-linear algebraic difference equations. We describe characteristic sets and compute difference dimension polynomials associated with some such systems. Then we apply our results to the comparative analysis of difference schemes for some PDEs from the point of view of their Einstein's strength. In particular, we determine the Einstein's strength of standard finite-difference schemes for the Murray, Burgers and some other reaction-diffusion equations.

参数化线性差分和微分方程的差分可积条件

Difference integrability conditions for parameterized linear difference and differential equations

本文研究了带差分参数的线性差分和微分方程组的可积性条件。证明了这样的系统是差分等单流的当且仅当它分别关于每个参数是差分等单流的。由于这一结果，不再需要求解非线性差分方程来验证同构性，这将提高这些系统的计算效率。

Algebraic groups as difference Galois groups of linear differential equations

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.

周期参数差分方程的伽罗瓦论

Galois theory of difference equations with periodic parameters

本文发展了周期参数线性差分方程系统的伽罗瓦论，并引入了线性差分代数群。然后，我们应用这个构造性地测试的解决方案的线性q-差分方程，与复杂的q，而不是一个单位根，满足任何多项式q '-差分方程与q'是一个单位根。