It is known that every solution to the second-order difference equation \(x_{n}=x_{n-1}+x_{n-2}=0\), \(n\ge 2\), can be written in the following form \(x_{n}=x_{0}f_{n-1}+x_{1}f_{n}\), where \(f_{n}\) is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

已知的是，每一个解决方案到第二阶差分方程\（X_ {N} = X_ {N-1} + X_ {N-2} = 0 \） ，\（N \ GE 2 \） ，可以写入格式为\（x_ {n} = x_ {0} f_ {n-1} + x_ {1} f_ {n} \），其中\（f_ {n} \）是斐波那契数列。在这里，我们找到所有具有任意阶数常数系数的齐次线性差分方程，这些方程的一般解具有相关形式的表示。我们还提出了一个有趣的基本过程，用于寻找具有常数系数的任何齐次线性差分方程的一般解的表示形式，该方程的系数，初始值和斐波那契数列的扩展都可以表示。当与方程相关联的特征多项式的所有根都互不相同的情况下，可以这样做，然后证明在其他情况下，这样获得的表示形式也成立。还显示出在应用程序期间，斐波那契序列的延伸自然地出现。