环上高阶线性差分方程的半共轭分解
Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings
摘要：
利用一种新的非线性方法研究环上变系数线性差分方程。在有单位元的幺环中，如果线性方程的齐次部分在环的单位群中有解（即，一个单一的解决方案），然后我们表明，该方程分解成两个线性方程的低阶。这种分解，在非线性理论中称为共轭分解，推广了线性背景下的经典算子分解。利用酉解的连续项之比序列来求其共轭因子分解。这样的序列，被称为特征序列，非常适合于变系数;例如，它们为经典的庞加莱-佩龙定理的表达提供了一个自然的背景。讨论了周期系数线性差分方程的一些应用，并导出了经典特殊函数如修正Bessel和Chebyshev所满足的线性泛函递推方程的通解公式.
英文摘要：
We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincaré-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev.
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