摘要：递推关系是离散变量之间变化规律中常见的一种方式。在数学上，递推关系也就是差分方程，它不仅在众多数学分支如组合、概率、几何、矩阵中起重要的作用，同时也在诸如信息科学领域中显示其独特的魅力。因此，对递推关系的探讨是一个相当具有实际意义的课题。

    本文首先简单论述线性递推关系的通解的定义，从而给出常系数齐次线性递推关系和常系数非齐次线性递推关系的通解的结构。其次介绍求解线性递推关系常用的迭代法、归纳法、特征根法、矩阵向量法和母函数法多种方法，并结合实例对各种方法的适用性予以辨析。最后用实例说明在概率、行列式、线性排列、染色等问题中递推关系的应用。

关键词：线性递推关系；母函数；递推关系；迭代法；特征向量；特征根法

Abstract: The recurrence relation is a common variation in discrete variables. Mathematically, the recurrence relation is the differential equation, which not only plays an important role in many branches of mathematics such as portfolio, probability, geometry, matrix, but also its unique charm, such as in the field of information science. Therefore, the exploration of the recurrence relation of a subject of considerable practical significance.

   In this paper,  we first simply discuss the general solution of linear recurrence relation definition, which gives constant coefficient linear recurrence relations with constant coefficients the general solution of the nonhomogeneous linear recurrence relation structure. Secondly, commonly used iterative method for solving linear recurrence relations, induction, eigenvalue method, a variety of methods of matrix-vector method and generating function method and two examples to be Analysis of the applicability of the various methods . Finally, an example to illustrate the application of probability, determinants, linear array, dyeing and other issues recurrence relations.

Key words:generating function; recurrence relations; eigenvalue method; particular solution; iterative method; eigenvector.