![TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]](Files/BellY.en/86.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.en/1.png "TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]"){kind=small}
Details
- Mathematical function, suitable for both symbolic and numeric manipulations.
- The partial Bell polynomial can be used to express the
derivative of a composition of two functions through the Faà di Bruno formula ![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.en/4.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]")
. » - The BellY polynomial
![TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]](Files/BellY.en/5.png "TemplateBox[{n, k, {x, _, 1}, ..., {x, _, {(, {n, -, k, +, 1}, )}}}, BellY]")
is given by 
⋯
Boole[m1+2 m2+⋯+n mnn∧m1+m2+⋯+mnk] 
(xs/s!)ms. » - The generalized Bell polynomial can be used to express the
derivative of a composition of 
functions ![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.en/13.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]")
. - BellY[n,k,m] is equivalent to BellY[
] where 
is formed by prepending UnitVector[n,k] to m as a first column. »
Background & Context
- BellY returns the partial Bell polynomial
of given variables 
or of a given matrix 
having 
rows, or the generalized Bell polynomial 
of any matrix 
. Here, 
, where the sum is over all sequences 
of non-negative integers satisfying 
and 
. This allows the partial Bell polynomial 
to encode how a set of 
objects can be partitioned into 
nonempty subsets
by summing together terms of the form 
, which indicates there are 
ways to partition the set of 
objects into 
subsets each of size 
, 
subsets each of size 
, etc. (so 
and 
necessarily). For instance, 
indicates that when partitioning a set of 6 objects into 2 nonempty subsets, there are 10 partitions consisting of 2 subsets each of size 3, 15 partitions consisting of 1 subset of size 2 and 1 subset of size 4, and 6 partitions consisting of 1 subset of size 1 and 1 subset of size 5. - Faà di Bruno's formula for generalizing the chain rule of differential calculus to express the
derivative of a composition of two functions is given in terms of the partial Bell polynomial by 
. The chain rule can be further generalized to express the 
derivative of a composition of 
functions using the generalized Bell polynomial 
, defined to be the unique function satisfying
(for example, 
by the ordinary chain rule). The generalized partial Bell polynomial 
is defined in terms of the generalized Bell polynomial 
by
![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.en/53.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]")
so that the generalized version of Faà di Bruno's formula given by
![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.en/54.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)...))")
holds. - Several famed integer sequences can be obtained by applying the partial Bell polynomial
to particular arguments. Letting ![TemplateBox[{n, k}, StirlingS1]](Files/BellY.en/56.png "TemplateBox[{n, k}, StirlingS1]")
and ![TemplateBox[{n, k}, StirlingS2]](Files/BellY.en/57.png "TemplateBox[{n, k}, StirlingS2]")
denote Stirling numbers of the first kind and second kind, respectively, given by StirlingS1 and StirlingS2, gives ![Y_(n,k)(0!,...,(n-k)!)=TemplateBox[{TemplateBox[{n, k}, StirlingS1, SyntaxForm -> SubsuperscriptBox]}, Abs]](Files/BellY.en/58.png "Y_(n,k)(0!,...,(n-k)!)=TemplateBox[{TemplateBox[{n, k}, StirlingS1, SyntaxForm -> SubsuperscriptBox]}, Abs]")
and ![Y_(n,k)(1,...,1)=TemplateBox[{n, k}, StirlingS2]](Files/BellY.en/59.png "Y_(n,k)(1,...,1)=TemplateBox[{n, k}, StirlingS2]")
. Furthermore, letting 
and 
respectively denote the 
Bell polynomial and 
Bell number, both given by BellB, gives ![TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x)](Files/BellY.en/66.png "TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x)")
and 
.
Examples
open all close allBasic Examples (3)
BellY[4, 2, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}]Generalized partial Bell polynomial:
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]}}]Applications (13)
Calculus (3)
The generalized chain rule allows one to directly compute the 
derivative of 
using 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.en/71.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}
. Verify this with symbolic 
and 
for low orders of 
:
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}]For 
, this becomes the normal chain rule:
GeneralizedChainRule[{f, g}, {x, 1}]D[f[g[x]], x]For 
, this is also known as Faà di Bruno's formula:
D[f[g[x]], {x, 2}]GeneralizedChainRule[{f, g}, {x, 2}]D[f[g[x]], {x, 3}]GeneralizedChainRule[{f, g}, {x, 3}]From the formula, it can be directly seen that the derivative is linear in 
with coefficients that are polynomial in 
, where ![TemplateBox[{n, k, {g, ^, {(, ', )}}, ..., {g, ^, {(, {(, {{-, k}, +, n, +, 1}, )}, )}}}, BellY]](Files/BellY.en/79.png "TemplateBox[{n, k, {g, ^, {(, ', )}}, ..., {g, ^, {(, {(, {{-, k}, +, n, +, 1}, )}, )}}}, BellY]"){kind=small}
is the polynomial coefficient of 
. Define a typesetting rule that makes this relation obvious by paneling the Bell coefficients:
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}], "+"]A few of the first derivatives:
ChainRuleTypeset[{f, g}, {x, 2}]ChainRuleTypeset[{f, g}, {x, 3}]ChainRuleTypeset[{f, g}, {x, 4}]ChainRuleTypeset[{f, g}, {x, 5}]Compute fourth-order derivatives of the Gamma function using the BellY polynomial:
Gamma[n]BellY[Table[{1, PolyGamma[k - 1, n]}, {k, 4}]]//ExpandCompare with the explicit evaluation of the derivative:
D[Gamma[n], {n, 4}]Compute the series of an inverse function:
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//SimplifyCompare with the result of InverseSeries:
InverseSeries[Underoverscript[∑, j = 1, n]C[j]x^j + O[x]^n + 1]//SimplifyCombinatorics (6)
Compute Stirling numbers of the first kind in terms of the partial Bell polynomials:
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}]Compute Stirling numbers of the second kind in terms of the partial Bell polynomials:
Table[BellY[n, k, ConstantArray[1, n]], {n, 0, 5}, {k, 0, n}]Table[StirlingS2[n, k], {n, 0, 5}, {k, 0, n}]Compute Bell numbers using generalized Bell polynomials:
Table[BellY[ConstantArray[1, {n, 2}]], {n, 10}]Table[BellB[n], {n, 10}]Compute Bell polynomials BellB[n,z] using generalized Bell polynomials:
Table[BellY[ConstantArray[{1, z}, n]], {n, 5}]Table[BellB[n, z], {n, 5}]Compute Catalan numbers using generalized Bell polynomials:
Table[BellY[Transpose[{1 / Range[n, 1, -1]!, Range[n]!}]], {n, 10}]Table[CatalanNumber[n], {n, 10}]Number of 
-level labeled rooted trees with 
leaves:
Table[BellY[ConstantArray[1, {n, n}]], {n, 1, 10}]Compare with an alternative formula:
sm = Table[StirlingS2[s, t], {s, 10}, {t, 10}];Table[MatrixPower[sm, n][[n, 1]], {n, 10}]The cycle index polynomial of the symmetric group of degree n:
n = 5;
vars = Table[Subscript[x, k], {k, n}];
BellY[Transpose[{ConstantArray[(1/n!), n], Range[0, n - 1]!vars}]]Compare with the result of CycleIndexPolynomial:
CycleIndexPolynomial[SymmetricGroup[n], vars]The cycle index polynomial of the alternating group of degree n:
BellY[Transpose[{Table[(1 + (-1)^n + k/n!), {k, n}], Range[0, n - 1]!vars}]]Compare with the result of CycleIndexPolynomial:
CycleIndexPolynomial[AlternatingGroup[n], vars]Find the number of ways to partition a set of 6 elements into two subsets from a partial Bell polynomial:
BellY[6, 2, Array[x, 6]]Check by explicit recursive generation of set partitions:
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]]There are 10 ways to partition a set of 6 elements into two subsets of 3+3 elements:
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {3, 3}]There are 15 ways to partition a set of 6 elements into two subsets of 4+2 elements:
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {2, 4}]There are 6 ways to partition a set of 6 elements into two subsets of 5+1 elements:
Count[SetPartitions[Range[6]], x_ /; Sort[Length /@ x] == {1, 5}]Other Applications (4)
Define the complete Bell polynomial of n variables:
bellComplete[vars__] := With[{tmp = {vars}}, BellY[Transpose[{ConstantArray[1, Length[tmp]], tmp}]]]Show the first few complete Bell polynomials:
{bellComplete[x], bellComplete[x, y], bellComplete[x, y, z], bellComplete[x, y, z, w]}//ColumnCompute the third raw moment in terms of cumulants:
BellY[Table[{1, Cumulant[k]}, {k, 3}]]MomentConvert[Moment[3], "Cumulant"]Compute the third cumulant in terms of raw moments:
BellY[Table[{(-1)^k - 1(k - 1)!, Moment[k]}, {k, 3}]]MomentConvert[Cumulant[3], "Moment"]Construct polynomial sequences of binomial type:
p[n_Integer ? NonNegative, x_] := Sum[BellY[n, k, Array[a, n]]x ^ k, {k, 0, n}]Verify their defining identity:
Table[p[n, x + y] == Sum[Binomial[n, k]p[k, x]p[n - k, y], {k, 0, n}]//Simplify, {n, 0, 10}]Recover BellB[n,z] as a special case:
Table[p[n, z], {n, 5}] /. a[_] -> 1Table[BellB[n, z], {n, 5}]The n
elementary symmetric polynomial can be defined in terms of BellY:
ee[n_, vars_] := BellY[Table[{(1/n!), (-1)^k - 1(k - 1)!Total[vars^k]}, {k, n}]]Compare with SymmetricPolynomial for the case of five variables:
Table[ee[k, {x, y, z, u, v}] == SymmetricPolynomial[k, {x, y, z, u, v}]//Simplify, {k, 5}]Properties & Relations (6)
Compute a partial Bell polynomial using its sum representation:
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]}]]Compare with BellY:
With[{n = 7, k = 2}, BellY[n, k, Array[, n]]]Compute a partial Bell polynomial using Cvijović's iterated sum formula:
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}]]]]Compare with BellY:
With[{n = 7, k = 2}, BellY[n, k, Array[, n]]]A linear combination of partial Bell polynomials:
n = 5;
Sum[C[k]BellY[n, k, Table[[k], {k, n}]], {k, n}]The equivalent expression in terms of the generalized Bell polynomial:
BellY[Table[{C[k], [k]}, {k, n}]]%% == %//SimplifyA generalized partial Bell polynomial of a matrix:
n = 4;k = 3;
mat = (| | |
| ------- | ------- |
| [1, 1] | [1, 2] |
| [2, 1] | [2, 2] |
| [3, 1] | [3, 2] |
| [4, 1] | [4, 2] |);
BellY[n, k, mat]This can be computed in terms of the generalized Bell polynomial by prepending a unit vector as a column:
BellY[Join[Transpose[{UnitVector[n, k]}], mat, 2]]A linear combination of generalized partial Bell polynomials of a matrix can be expressed as a generalized Bell polynomial by prepending the column of coefficients to the matrix:
Sum[C[k]BellY[n, k, mat], {k, n}] == BellY[Join[Transpose[{Table[C[k], {k, n}]}], mat, 2]]//SimplifyFaà di Bruno's formula for the third derivative of 
:
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}]Demonstrate an inversion relation for generalized Bell polynomials:
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}]//SimplifyNeat Examples (2)
Generate Bernoulli numbers using a generalized Bell polynomial:
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}]Generate Euler numbers using a generalized Bell polynomial:
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}]Text
Wolfram Research (2010), BellY, Wolfram Language function, https://reference.wolfram.com/language/ref/BellY.html.
CMS
Wolfram Language. 2010. "BellY." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellY.html.
APA
Wolfram Language. (2010). BellY. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellY.html