DifferenceRootReduce
DifferenceRootReduce[expr,n]
尽可能地将 expr 化简成单个关于 n 的函数 DifferenceRoot 对象.
更多信息和选项
- DifferenceRootReduce 尽可能地将任何表达式表示为一个 DifferenceRoot 对象.
- 当关于 expr 的 DifferenceRoot 对象等价于零序列时，DifferenceRootReduce[expr,n] 总是给出 0.
- DifferenceRootReduce 自动逐项作用于列表、方程和不等式的元素.
- DifferenceRootReduce[f] 作用于纯函数或纯 DifferenceRoot 对象.
- DifferenceRootReduce 接受以下选项：
-
Assumptions $Assumptions 关于参数的假设 Method Automatic 使用的方法
范例
打开所有单元 关闭所有单元基本范例 (2)
将斐波那契 (Fibonacci) 数列简化为 DifferenceRoot 对象：
DifferenceRootReduce[Fibonacci[k], k]Table[%, {k, 0, 10}]Table[Fibonacci[k], {k, 0, 10}]DifferenceRootReduce[Fibonacci[k + 1]Fibonacci[k - 1] - Fibonacci[k] ^ 2 == (-1) ^ k, k]范围 (9)
DifferenceRootReduce[n ^ 2 + 1, n]DifferenceRootReduce[(n + 1) / (n - 1), n]DifferenceRootReduce[Binomial[2n, n], n]DifferenceRootReduce[2 ^ #&]Verify the DifferenceRoot object:
Table[DifferentialRoot[Function[{, }, {(-[])*Log[2] + Derivative[1][][] == 0,
[0] == 1}]][n], {n, 10}]DifferenceRootReduce[Fibonacci[n] + Exp[n], n]DifferenceRootReduce[Binomial[2n, n]Exp[n], n]DifferenceRootReduce[Fibonacci[2n + 1], n]DifferenceRootReduce[Fibonacci[n] - (-1) ^ n LucasL[2n + 1], n]DifferentialRootReduce 自动逐项作用于列表的各个元素：
DifferenceRootReduce[{LucasL[n], n}, n]选项 (2)
Assumptions (1)
DifferenceRootReduce 可以根据关于变量的假设产生不同的循环结构：
expr = (-1)^Sqrt[n^2];DifferenceRootReduce 在一般情况下将表达式保持在未经运算的状态：
DifferenceRootReduce[expr, n]在假设 
的情况下，DifferenceRootReduce 会生成该表达式的二阶差分方程：
DifferenceRootReduce[expr, n, Assumptions -> n ≥ 0]Method (1)
DifferentialRootReduce 可以给出非齐次方程：
DifferenceRootReduce[Fibonacci[n] + 1, n]用选项 Method->"Homogeneous" 得到齐次方程：
DifferenceRootReduce[Fibonacci[n] + 1, n, Method -> "Homogeneous"]应用 (3)
验证 Fibonacci 的取负公式：
DifferenceRootReduce[Fibonacci[-n] == (-1) ^ (n + 1) * Fibonacci[n], n]DifferenceRootReduce[Fibonacci[m + n] == 1 / 2(Fibonacci[m]LucasL[n] + LucasL[m]Fibonacci[n]), n]DifferenceRootReduce[Fibonacci[2n] == Fibonacci[n]LucasL[n], n]DifferenceRootReduce[Fibonacci[n + 1] == 1 / 2(Fibonacci[n] + LucasL[n]), n]DifferenceRootReduce[Fibonacci[n] == 1 / 5(LucasL[n - 1] + LucasL[n + 1]), n]HoldForm[Sum[Binomial[2m + 1, 2k]3 ^ k / (5 2 ^ m), {k, 0, m}]]
DifferenceRootReduce[Sum[Binomial[2m + 1, 2k]3 ^ k / (5 2 ^ m), {k, 0, m}] /. m -> 3n + 1, n]Table[%, {n, 0, 10}]//SimplifyPadovanNumber = DifferenceRoot[Function[{y, n}, {-y[n] - y[1 + n] + y[3 + n] == 0, y[0] == 1, y[1] == 1, y[2] == 1}]]Table[PadovanNumber[n], {n, 0, 10}]DifferenceRootReduce[PadovanNumber[n] == PadovanNumber[n - 2] + PadovanNumber[n - 3], n]DifferenceRootReduce[PadovanNumber[n] == PadovanNumber[n - 4] + PadovanNumber[n - 5] + PadovanNumber[n - 6] + PadovanNumber[n - 7] + PadovanNumber[n - 8], n]DifferenceRootReduce[Sum[PadovanNumber[3k + 2], {k, 0, n}] == PadovanNumber[3n + 4] - 1, n]文本
Wolfram Research (2008)，DifferenceRootReduce，Wolfram 语言函数，https://reference.wolfram.com/language/ref/DifferenceRootReduce.html （更新于 2020 年）.
CMS
Wolfram 语言. 2008. "DifferenceRootReduce." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2020. https://reference.wolfram.com/language/ref/DifferenceRootReduce.html.
APA
Wolfram 语言. (2008). DifferenceRootReduce. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/DifferenceRootReduce.html 年