Abstract In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation x n + 1 = a ⁢ x n ⁢ x n - k b ⁢ x n - c ⁢ x n - k - 1 , n = 0 , 1 , … , x_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots, where a , b , c {a,b,c} are positive real numbers and the initial conditions x - k - 1 , x - k , … , x - 1 , x 0 {x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}} are real numbers. We show that when a = b = c {a=b=c} , the behavior of the solutions depends on whether k is even or odd.

中文翻译：

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关于高阶差分方程的解

摘要 在本文中，我们确定了禁集，引入了解的显式公式，并讨论了差分方程 xn + 1 = a ⁢ xn ⁢ xn - kb ⁢ xn - c ⁢ xn - k - 1 的解的全局行为, n = 0 , 1 , … , x_{n+1}=\frac{ax_{n}x_{nk}}{bx_{n}-cx_{nk-1}},\quad n=0,1, \ldots，其中 a , b , c {a,b,c} 是正实数和初始条件 x - k - 1 , x - k , … , x - 1 , x 0 {x_{-k-1} ,x_{-k},\ldots,x_{-1},x_{0}} 是实数。我们表明，当 a = b = c {a=b=c} 时，解的行为取决于 k 是偶数还是奇数。