BellY
![TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]](Files/BellY.zh/30.png "TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]"){kind=small}
BellY
![TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]](Files/BellY.zh/1.png "TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]"){kind=small}
![D_t^nf(g(t))=sum_(k=0)^nf^((k))(g(t)) TemplateBox[{n, k, {{g, ^, {(, ', )}}, (, t, )}, {{g, ^, {(, '', )}}, (, t, )}, ..., {{g, ^, {(, {(, {n, -, k, +, 1}, )}, )}}, (, t, )}}, BellY]](Files/BellY.zh/2.png "D_t^nf(g(t))=sum_(k=0)^nf^((k))(g(t)) TemplateBox[{n, k, {{g, ^, {(, ', )}}, (, t, )}, {{g, ^, {(, '', )}}, (, t, )}, ..., {{g, ^, {(, {(, {n, -, k, +, 1}, )}, )}}, (, t, )}}, BellY]"){kind=small}
阶导数. ![TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]](Files/BellY.zh/4.png "TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]"){kind=small}
阶导数![D_t^nf_1(f_2(...f_m(t)...))⩵TemplateBox[{{{{{{f, _, 1}, '}, {(, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , {{{f, _, 2}, '}, {(, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , ..., , {{{f, _, m}, '}, {(, t, )}}}, ; , {{{{f, _, 1}, ''}, {(, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , {{{f, _, 2}, ''}, {(, {{f, _, 3}, {(, {..., , {{f, _, m}, (, t, )}, ...}, )}}, )}}, , ..., , {{{f, _, m}, ''}, {(, t, )}}}, ; , {|, , |, , ..., , |}, ; , {{{{f, _, 1}, ^, {(, {(, n, )}, )}}, (, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , {{{f, _, 2}, ^, {(, {(, n, )}, )}}, (, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , ..., , {{{f, _, m}, ^, {(, {(, n, )}, )}}, (, t, )}}}}, BellY1]](Files/BellY.zh/10.png "D_t^nf_1(f_2(...f_m(t)...))⩵TemplateBox[{{{{{{f, _, 1}, '}, {(, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , {{{f, _, 2}, '}, {(, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , ..., , {{{f, _, m}, '}, {(, t, )}}}, ; , {{{{f, _, 1}, ''}, {(, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , {{{f, _, 2}, ''}, {(, {{f, _, 3}, {(, {..., , {{f, _, m}, (, t, )}, ...}, )}}, )}}, , ..., , {{{f, _, m}, ''}, {(, t, )}}}, ; , {|, , |, , ..., , |}, ; , {{{{f, _, 1}, ^, {(, {(, n, )}, )}}, (, {{f, _, 2}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , {{{f, _, 2}, ^, {(, {(, n, )}, )}}, (, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , ..., , {{{f, _, m}, ^, {(, {(, n, )}, )}}, (, t, )}}}}, BellY1]")
背景
- BellY 返回部分贝尔多项式
，其变量为 
，或者给定矩阵 
的部分贝尔多项式，它有 
行，或广义贝尔多项式 
，适用于任意矩阵 
. 这里，
，其中求和遍历所有满足条件的序列 
，它们是满足 
且 
的非负整数序列. 由此，部分贝尔多项式 
能够表征将一个含有 
个对象的集合如何划分为 
个非空子集的所有可能方式
，其通过对形如 
的项进行求和来实现，其中每一种表示有 
种方式将含有 
个对象的集合划分为 
个大小为 
的子集，
个大小为 
的子集，依此类推（故 
且 
是必须满足的条件）. 例如，
表明当把一个 6 个对象的集合划分为 2 个非空子集时，存在 10 种划分方式形成两个大小为 3 的子集，15 种划分方式形成一个大小为 2 的子集和一个大小为 4 的子集，以及 6 种划分形成一个大小为 1 的子集和一个大小为 5 的子集. - 用于推广微分链式法则的 Faà di Bruno 公式，通过部分贝尔多项式将两函数复合的
阶导数表述为：
. 该链式法则可以进一步推广至 
个函数复合的情形，此时需借助广义贝尔多项式 
，其定义为满足以下条件的唯一函数：
（例如，
就是通过通常的链式法则得到的）. 广义偏贝尔多项式 
可以通过广义贝尔多项式 
来定义：
![TemplateBox[{n, k, {{{z, _, {(, {1, ,, 1}, )}}, , {z, _, {(, {1, ,, 2}, )}}, , ..., , {z, _, {(, {1, ,, m}, )}}}, ; , {|, , |, , |, , |}, ; , {{z, _, {(, {k, ,, 1}, )}}, , {z, _, {(, {k, ,, 2}, )}}, , ..., , {z, _, {(, {k, ,, m}, )}}}, ; , {|, , |, , |, , |}, ; , {{z, _, {(, {n, ,, 1}, )}}, , {z, _, {(, {n, ,, 2}, )}}, , ..., , {z, _, {(, {n, ,, m}, )}}}}}, BellYMatrix]=TemplateBox[{{{0, , {z, _, {(, {1, ,, 1}, )}}, , {z, _, {(, {1, ,, 2}, )}}, , ..., , {z, _, {(, {1, ,, m}, )}}}, ; , {|, , |, , |, , |, , |}, ; , {0, , {z, _, {(, {{k, -, 1}, ,, 1}, )}}, , {z, _, {(, {{k, -, 1}, ,, 2}, )}}, , ..., , {z, _, {(, {{k, -, 1}, ,, m}, )}}}, ; , {1, , {z, _, {(, {k, ,, 1}, )}}, , {z, _, {(, {k, ,, 2}, )}}, , ..., , {z, _, {(, {k, ,, m}, )}}}, ; , {0, , {z, _, {(, {{k, +, 1}, ,, 1}, )}}, , {z, _, {(, {{k, +, 1}, ,, 2}, )}}, , ..., , {z, _, {(, {{k, +, 1}, ,, m}, )}}}, ; , {|, , |, , |, , |, , |}, ; , {0, , {z, _, {(, {n, ,, 1}, )}}, , {z, _, {(, {n, ,, 2}, )}}, , ..., , {z, _, {(, {n, ,, m}, )}}}}}, BellY1]](Files/BellY.zh/17.png "TemplateBox[{n, k, {{{z, _, {(, {1, ,, 1}, )}}, , {z, _, {(, {1, ,, 2}, )}}, , ..., , {z, _, {(, {1, ,, m}, )}}}, ; , {|, , |, , |, , |}, ; , {{z, _, {(, {k, ,, 1}, )}}, , {z, _, {(, {k, ,, 2}, )}}, , ..., , {z, _, {(, {k, ,, m}, )}}}, ; , {|, , |, , |, , |}, ; , {{z, _, {(, {n, ,, 1}, )}}, , {z, _, {(, {n, ,, 2}, )}}, , ..., , {z, _, {(, {n, ,, m}, )}}}}}, BellYMatrix]=TemplateBox[{{{0, , {z, _, {(, {1, ,, 1}, )}}, , {z, _, {(, {1, ,, 2}, )}}, , ..., , {z, _, {(, {1, ,, m}, )}}}, ; , {|, , |, , |, , |, , |}, ; , {0, , {z, _, {(, {{k, -, 1}, ,, 1}, )}}, , {z, _, {(, {{k, -, 1}, ,, 2}, )}}, , ..., , {z, _, {(, {{k, -, 1}, ,, m}, )}}}, ; , {1, , {z, _, {(, {k, ,, 1}, )}}, , {z, _, {(, {k, ,, 2}, )}}, , ..., , {z, _, {(, {k, ,, m}, )}}}, ; , {0, , {z, _, {(, {{k, +, 1}, ,, 1}, )}}, , {z, _, {(, {{k, +, 1}, ,, 2}, )}}, , ..., , {z, _, {(, {{k, +, 1}, ,, m}, )}}}, ; , {|, , |, , |, , |, , |}, ; , {0, , {z, _, {(, {n, ,, 1}, )}}, , {z, _, {(, {n, ,, 2}, )}}, , ..., , {z, _, {(, {n, ,, m}, )}}}}}, BellY1]")
因此，Faà di Bruno 公式的广义形式可写为：
![D_t^nf_1(f_2(...f_m(t)...))=sum_(k=1)^nTemplateBox[{n, k, {{{{{f, _, 2}, '}, {(, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , ..., , {{{f, _, m}, '}, {(, t, )}}}, ; , {{{{f, _, 2}, ''}, {(, {{f, _, 3}, {(, {..., , {{f, _, m}, (, t, )}, ...}, )}}, )}}, , ..., , {{{f, _, m}, ''}, {(, t, )}}}, ; , {|, , |, , |}, ; , {{{{f, _, 2}, ^, {(, {(, n, )}, )}}, (, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , ..., , {{{f, _, m}, ^, {(, {(, n, )}, )}}, (, t, )}}}}, BellYMatrix]f_1^((k))(f_2(... f_m(t)...))](Files/BellY.zh/18.png "D_t^nf_1(f_2(...f_m(t)...))=sum_(k=1)^nTemplateBox[{n, k, {{{{{f, _, 2}, '}, {(, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}}, , ..., , {{{f, _, m}, '}, {(, t, )}}}, ; , {{{{f, _, 2}, ''}, {(, {{f, _, 3}, {(, {..., , {{f, _, m}, (, t, )}, ...}, )}}, )}}, , ..., , {{{f, _, m}, ''}, {(, t, )}}}, ; , {|, , |, , |}, ; , {{{{f, _, 2}, ^, {(, {(, n, )}, )}}, (, {{f, _, 3}, (, {..., , {{f, _, m}, (, t, )}, ...}, )}, )}, , ..., , {{{f, _, m}, ^, {(, {(, n, )}, )}}, (, t, )}}}}, BellYMatrix]f_1^((k))(f_2(... f_m(t)...))")
成立. - 一些著名的整数数列可以通过将部分贝尔多项式
应用于特定的参数而得到. 设 ![TemplateBox[{n, k}, StirlingS1]](Files/BellY.zh/19.png "TemplateBox[{n, k}, StirlingS1]")
和 ![TemplateBox[{n, k}, StirlingS2]](Files/BellY.zh/20.png "TemplateBox[{n, k}, StirlingS2]")
分别表示第一类及第二类斯特林数，其由 StirlingS1 与 StirlingS2 给出，则有 ![Y_(n,k)(0!,...,(n-k)!)=TemplateBox[{TemplateBox[{n, k}, StirlingS1, SyntaxForm -> SubsuperscriptBox]}, Abs]](Files/BellY.zh/21.png "Y_(n,k)(0!,...,(n-k)!)=TemplateBox[{TemplateBox[{n, k}, StirlingS1, SyntaxForm -> SubsuperscriptBox]}, Abs]")
与 ![Y_(n,k)(1,...,1)=TemplateBox[{n, k}, StirlingS2]](Files/BellY.zh/22.png "Y_(n,k)(1,...,1)=TemplateBox[{n, k}, StirlingS2]")
. 进一步地，设 
和 
分别表示第 
个贝尔多项式和第 
个贝尔数，它们由 BellB 给出，则有 ![TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x)](Files/BellY.zh/25.png "TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x)")
与 
.
范例
打开所有单元 关闭所有单元基本范例 (3)
BellY[4, 2, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}]BellY[4, 2, {{Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[y, 1], Subscript[y, 2], Subscript[y, 3]}}]BellY[{{Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[y, 1], Subscript[y, 2], Subscript[y, 3]}, {Subscript[z, 1], Subscript[z, 2], Subscript[z, 3]}}]应用 (13)
微积分 (3)
广义链式法则允许使用 BellY:![(partial^nf(g(x)))/(partialx^n)=sum_(k=1)^nf^((k))(g(x)) TemplateBox[{n, k, {{(, {partial, {g, (, x, )}}, )}, /, {(, {partial, {x, ^, 1}}, )}}, ..., {{(, {{partial, ^, {(, {n, -, k, +, 1}, )}}, {g, (, x, )}}, )}, /, {(, {partial, {x, ^, {(, {n, -, k, +, 1}, )}}}, )}}}, BellY]](Files/BellY.zh/26.png "(partial^nf(g(x)))/(partialx^n)=sum_(k=1)^nf^((k))(g(x)) TemplateBox[{n, k, {{(, {partial, {g, (, x, )}}, )}, /, {(, {partial, {x, ^, 1}}, )}}, ..., {{(, {{partial, ^, {(, {n, -, k, +, 1}, )}}, {g, (, x, )}}, )}, /, {(, {partial, {x, ^, {(, {n, -, k, +, 1}, )}}}, )}}}, BellY]"){kind=small}
直接计算 
的 
阶导数，对于 
的较低阶数和符号式 
和 
验证这一点：
GeneralizedChainRule[{f_, g_}, {x_, n_}] :=
Sum[Derivative[k][f][g[x]]BellY[n, k, Table[D[g[x], {x, i}], {i, n - k + 1}]], {k, n}]
GeneralizedChainRule[{f, g}, {x, 1}]D[f[g[x]], x]
D[f[g[x]], {x, 2}]GeneralizedChainRule[{f, g}, {x, 2}]D[f[g[x]], {x, 3}]GeneralizedChainRule[{f, g}, {x, 3}]根据此式，可以直接看出，对于 
中的多项式系数，
中的导数是线性的，其中 ![TemplateBox[{n, k, {g, ^, {(, ', )}}, ..., {g, ^, {(, {(, {{-, k}, +, n, +, 1}, )}, )}}}, BellY]](Files/BellY.zh/27.png "TemplateBox[{n, k, {g, ^, {(, ', )}}, ..., {g, ^, {(, {(, {{-, k}, +, n, +, 1}, )}, )}}}, BellY]"){kind=small}
是 
的多项式系数. 定义一个排版格式，通过镶嵌贝尔系数，使得这种关系是显而易见的：
pnl[e_Plus, col_] := Panel[Row[{"(", e, ")"}], Background -> col];
pnl[e_, col_] := Panel[e, Background -> col];
ChainRuleTypeset[{f_, g_}, {x_, n_}] := Row[Table[Row[{pnl[BellY[n, k, Table[D[g[x], {x, i}], {i, n - k + 1}]], StandardBlue], pnl[Derivative[k][f][g[x]], StandardOrange]}, " "], {k, n}], "+"]ChainRuleTypeset[{f, g}, {x, 2}]ChainRuleTypeset[{f, g}, {x, 3}]ChainRuleTypeset[{f, g}, {x, 4}]ChainRuleTypeset[{f, g}, {x, 5}]利用 BellY 多项式计算 Gamma 函数的 4 阶导数：
Gamma[n]BellY[Table[{1, PolyGamma[k - 1, n]}, {k, 4}]]//ExpandD[Gamma[n], {n, 4}]n = 4;
(x/C[1]) + Underoverscript[∑, j = 2, n](x^j/C[1]^jj!)BellY[Table[{Pochhammer[j, k - 1], -(k - 1)! (C[k]/C[1])}, {k, 2, j}]] + O[x]^n + 1//Simplify与 InverseSeries 的结果比较：
InverseSeries[Underoverscript[∑, j = 1, n]C[j]x^j + O[x]^n + 1]//Simplify组合数学 (6)
Table[BellY[n, k, Table[(-1)^jj!, {j, 0, n}]], {n, 0, 5}, {k, 0, n}]Table[StirlingS1[n, k], {n, 0, 5}, {k, 0, n}]Table[BellY[n, k, ConstantArray[1, n]], {n, 0, 5}, {k, 0, n}]Table[StirlingS2[n, k], {n, 0, 5}, {k, 0, n}]Table[BellY[ConstantArray[1, {n, 2}]], {n, 10}]Table[BellB[n], {n, 10}]使用广义贝尔多项式计算贝尔多项式 BellB[n,z]：
Table[BellY[ConstantArray[{1, z}, n]], {n, 5}]Table[BellB[n, z], {n, 5}]Table[BellY[Transpose[{1 / Range[n, 1, -1]!, Range[n]!}]], {n, 10}]Table[CatalanNumber[n], {n, 10}]
Table[BellY[ConstantArray[1, {n, n}]], {n, 1, 10}]sm = Table[StirlingS2[s, t], {s, 10}, {t, 10}];Table[MatrixPower[sm, n][[n, 1]], {n, 10}]n = 5;
vars = Table[Subscript[x, k], {k, n}];
BellY[Transpose[{ConstantArray[(1/n!), n], Range[0, n - 1]!vars}]]与 CycleIndexPolynomial 的结果对比：
CycleIndexPolynomial[SymmetricGroup[n], vars]BellY[Transpose[{Table[(1 + (-1)^n + k/n!), {k, n}], Range[0, n - 1]!vars}]]与 CycleIndexPolynomial 的结果对比：
CycleIndexPolynomial[AlternatingGroup[n], vars]用贝尔局部多项式求得将一个包含 6 个元素的集合分成两个子集的方法数量：
BellY[6, 2, Array[x, 6]]AdjoinElement[s_, el_] := Table[Sequence@@Join[Table[Insert[o, el, {k, -1}], {k, Length[o]}], {Insert[o, {el}, -1]}], {o, s}]SetPartitions[{}] := {{}};
SetPartitions[s_List] := AdjoinElement[SetPartitions[Most[s]], Last[s]]有 10 种方法可以将一个包含 6 个元素的集合分成两个 3+3 个元素的子集：
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {3, 3}]有 15 种方法可以将一个包含 6 个元素的集合分成两个 4+2 个元素的子集：
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {2, 4}]有 6 种方法可以将一个包含 6 个元素的集合分成两个 5+1 个元素的子集：
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {1, 5}]其他应用 (4)
bellComplete[vars__] := With[{tmp = {vars}}, BellY[Transpose[{ConstantArray[1, Length[tmp]], tmp}]]]{bellComplete[x], bellComplete[x, y], bellComplete[x, y, z], bellComplete[x, y, z, w]}//ColumnBellY[Table[{1, Cumulant[k]}, {k, 3}]]MomentConvert[Moment[3], "Cumulant"]BellY[Table[{(-1)^k - 1(k - 1)!, Moment[k]}, {k, 3}]]MomentConvert[Cumulant[3], "Moment"]p[n_Integer ? NonNegative, x_] := Sum[BellY[n, k, Array[a, n]]x ^ k, {k, 0, n}]Table[p[n, x + y] == Sum[Binomial[n, k]p[k, x]p[n - k, y], {k, 0, n}]//Simplify, {n, 0, 10}]将 BellB[n,z] 作为特例复原：
Table[p[n, z], {n, 5}] /. a[_] -> 1Table[BellB[n, z], {n, 5}]第 n
个基本对称多项式可以用 BellY 定义：
ee[n_, vars_] := BellY[Table[{(1/n!), (-1)^k - 1(k - 1)!Total[vars^k]}, {k, n}]]在五个变量的情况下，与 SymmetricPolynomial 进行比较：
Table[ee[k, {x, y, z, u, v}] == SymmetricPolynomial[k, {x, y, z, u, v}]//Simplify, {k, 5}]属性和关系 (6)
With[{n = 7, k = 2},
Sum[n!Apply[Times, ((Array[, n]/Range[n]!))^m(1/m!)], {m, FrobeniusSolve[Range[n], n]⋂FrobeniusSolve[ConstantArray[1, n], k]}]]与 BellY 比较：
With[{n = 7, k = 2}, BellY[n, k, Array[, n]]]With[{n = 7, k = 2},
Sum[(Binomial[n, i[1]]/k!)[n - i[1]][i[k - 1]]Underoverscript[∏, r = 1, k - 2]Binomial[i[r], i[r + 1]][i[r] - i[r + 1]], Evaluate[Sequence@@Prepend[Table[{i[r + 1], k - r - 1, i[r] - 1}, {r, k - 2}], {i[1], k - 1, n - 1}]]]]与 BellY 进行比较：
With[{n = 7, k = 2}, BellY[n, k, Array[, n]]]n = 5;
Sum[C[k]BellY[n, k, Table[[k], {k, n}]], {k, n}]BellY[Table[{C[k], [k]}, {k, n}]]%% == %//Simplifyn = 4;k = 3;
mat = (| | |
| ------- | ------- |
| [1, 1] | [1, 2] |
| [2, 1] | [2, 2] |
| [3, 1] | [3, 2] |
| [4, 1] | [4, 2] |);
BellY[n, k, mat]这可以用广义贝尔多项式来计算，方法是加入一个单位向量作为列：
BellY[Join[Transpose[{UnitVector[n, k]}], mat, 2]]矩阵的广义部分贝尔多项式的线性组合，可以通过在矩阵中加入系数列表达为广义贝尔多项式：
Sum[C[k]BellY[n, k, mat], {k, n}] == BellY[Join[Transpose[{Table[C[k], {k, n}]}], mat, 2]]//Simplify
Sum[Derivative[k][f][g[z]] BellY[3, k, {Derivative[1][g][z], Derivative[2][g][z], Derivative[3][g][z]}], {k, 1, 3}]BellY[{{Derivative[1][f][g[z]], Derivative[1][g][z]}, {Derivative[2][f][g[z]], Derivative[2][g][z]}, {Derivative[3][f][g[z]], Derivative[3][g][z]}}]D[f[g[z]], {z, 3}]n = 5;
yk = Table[BellY[Table[{1, [j]}, {j, k}]], {k, n}]Table[BellY[Table[{(-1)^j - 1(j - 1)!, Subscript[yk, [[j]]]}, {j, k}]], {k, n}]//Simplify巧妙范例 (2)
SeriesCoefficient[(Exp[t] - 1/t), {t, 0, r}, Assumptions -> r ≥ 0]r!SeriesCoefficient[u^-1, {u, 1, r}, Assumptions -> r ≥ 0]r!Table[BellY[Table[{%, %%}, {r, n}]], {n, 20}]Table[BernoulliB[n], {n, 20}]SeriesCoefficient[Cosh[t], {t, 0, r}, Assumptions -> r ≥ 0]r!//SimplifySeriesCoefficient[u^-1, {u, 1, r}, Assumptions -> r ≥ 0]r!Table[BellY[Table[{%, %%}, {r, n}]], {n, 20}]Table[EulerE[n], {n, 20}]
Wolfram Research (2010)，BellY，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BellY.html.
文本
Wolfram Research (2010)，BellY，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BellY.html.
CMS
Wolfram 语言. 2010. "BellY." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/BellY.html.
APA
Wolfram 语言. (2010). BellY. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/BellY.html 年