BellY

BellY[n,k,{x1,,xn-k+1}]

gives the partial Bell polynomial .

BellY[n,k,m]

gives the generalized partial Bell polynomial of a matrix m.

BellY[m]

gives the generalized Bell polynomial of a matrix m.

Details

  • Mathematical function, suitable for both symbolic and numeric manipulations.
  • The BellY polynomial is given by Boole[m1+2 m2++n mnnm1+m2++mnk] (xs/s!)ms.
  • The partial Bell polynomial can be used to express the ^(th) derivative of a composition of two functions through the Faà di Bruno formula .
  • The generalized Bell polynomial can be used to express the ^(th) derivative of a composition of functions .
  • BellY[n,k,m] is equivalent to BellY[] where is formed by adding UnitVector[n,k] to m as a first column.

Examples

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Basic Examples  (3)

Partial Bell polynomial:

Generalized partial Bell polynomial:

Generalized Bell polynomial:

Scope  (1)

Evaluate for numerical matrix:

Applications  (8)

The generalized chain rule allows one to directly compute the ^(th) 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]. Verify this for symbolic and for low orders of :

For , this becomes the normal chain rule:

For , this is also known as Faà di Bruno's formula:

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] is the polynomial coefficient of . Define a typesetting that makes this relation be obvious by paneling the Bell coefficients:

A few of the first derivatives:

Compute the third raw moment in terms of cumulants:

Compute the third cumulant in terms of raw moments:

Compute a partial Bell polynomial using sum representation:

Compare with BellY:

Number of -level labeled rooted trees with leaves:

Alternatively:

Compute fourth-order derivatives of the Gamma function using the BellY polynomial:

Compute the series of an inverse function:

Find the number of ways to partition a set of 6 elements into two subsets from a partial Bell polynomial:

Check by explicit recursive generation of set partitions:

There are 10 ways to partition a set of 6 elements into two subsets of 3+3 elements:

There are 15 ways to partition a set of 6 elements into two subsets of 4+2 elements:

There are 6 ways to partition a set of 6 elements into two subsets of 5+1 elements:

Construct polynomial sequences of binomial type:

Verify their defining identity:

Recover BellB[n,z] as a special case:

Properties & Relations  (3)

Faà di Bruno's formula for the third derivative of :

Stirling numbers of the second kind:

Compute Bell numbers using generalized Bell polynomials:

Neat Examples  (1)

Generate Bernoulli numbers using a generalized Bell polynomial:

Introduced in 2010
 (8.0)