常系数线性差分方程通解的显式表示

THE EXPLICIT REPRESENTATION OF GENERAL SOLUTION OF LINEAR DIFFERENCE EQUATION WITH CONSTANT COEFFICIENTS

将 n阶常系数线性差分方程化为矩阵形式 ,利用 Frobenius矩阵求得特解的显式表示 ,进而求得它的通解的显式表示 ,该通解不涉及不定方程求非负解问题 ,并给出用二项式系数表示广义 Fibonacci数列解的关系式 .

This paper turns n th order linear difference equation with constant coefficients into the matrix form, gets the explicit representation of special solution of the equation via Frobenius matrix and obtains the explicit representation of general solution of the equation. Compared with the usual approach, the method given in this paper is independent of solving nonnegative integer solutions of the linear diophantine equation. It also obtains the solution of generalized Fibonacci sequence, which is made up by binomial coefficients.