The BellB, IncompleteBellB, and CompleteBellB respectively represent the Bell polynomials, the incomplete Bell polynomials - also called Bell polynomials of the second kind - and the complete Bell polynomials. For the Bell numbers, see bell.

The BellB polynomials are polynomials of degree $n$ defined in terms of the Stirling numbers of the second kind as

For the definition of the IncompleteBellB polynomials, consider a sequence $z_n$ with $n=1,2,3,\mathrm{...}$, with which we construct the sequence

where the $n^{\mathrm{th}}$ element is here defined as

Taking $z\diamond z\diamond z=\left(\left(z\diamond z\right)\diamond z\right)$, the IncompleteBellB polynomials are defined in terms of an operation $\left(z\diamond ...\diamond z\right)$ involving $k$ factors as

The output of IncompleteBellB is thus a multivariable polynomial of degree $k$ in the $z_j$ variables. Note that the right-hand side of this formula involves only the first $n-k+1$ elements of the sequence $z_j$; so in the left-hand side only the first $n-k+1$ $z_j$ are relevant, and all those not given in the input to IncompleteBellB will be assumed equal to zero.

To compute the first $n$ elements of the sequence obtained by performing this diamond operation $\left(z\diamond ...\diamond z\right)$ between $k$ factors you can use the IncompleteBellB:-DiamondConvolution command. This command makes use of the first $n-k+1$ elements of the sequence $z_j$ and returns a sequence of $n$ elements, where the first $k-1$ are equal to zero and the remaining $n-k+1$ are all polynomials of degree $k$ in the $z_j$ variables. Note that, unlike IncompleteBellB, IncompleteBellB:-DiamondConvolution expects the sequence $z_j$ enclosed as a list as third argument (see the Examples section).

The CompleteBellB polynomials are in turn defined in terms of the IncompleteBellB polynomials as

When the sequence $z_j$ passed to CompleteBellB contains less than $n$ elements, the missing ones will be assumed equal to zero.

All of CompleteBellB, IncompleteBellB and IncompleteBellB:-DiamondConvolution accept inert sequences constructed with %seq or the quoted 'seq' functions as part of the $z_j$ arguments, in which case they return unevaluated, echoing the input.

The Bell polynomials appear in various applications, including for instance Faà di Bruno's formula

where $f^{\left(k\right)}\left(g\left(x\right)\right)$ represents the $k^{\mathrm{th}}$ derivative of $f\left(x\right)$ evaluated at $g\left(x\right)$; the exponential of a formal power series

and in the following exponential generating function

$\mathrm{BellB}\left(n\,z\right)=\mathrm{Sum}\left(\mathrm{Stirling2}\left(n\,k\right)z^k\,k=0..n\right)$

$\mathrm{BellB}\left(n\,z\right)=\sum _{k=0}^n\mathrm{Stirling2}\left(n\,k\right)z^k$

$\mathrm{eval}\left(\,n=4\right)$

$z^4+6z^3+7z^2+z=\sum _{k=0}^4\mathrm{Stirling2}\left(4\,k\right)z^k$

$\mathrm{value}\left(\right)$

$z^4+6z^3+7z^2+z=z^4+6z^3+7z^2+z$

$\mathrm{seq}\left('\mathrm{BellB}'\left(n\,z\right)=\mathrm{BellB}\left(n\,z\right)\,n=0..3\right)$

$\mathrm{BellB}\left(0\,z\right)=1,\mathrm{BellB}\left(1\,z\right)=z,\mathrm{BellB}\left(2\,z\right)=z^2+z,\mathrm{BellB}\left(3\,z\right)=z^3+3z^2+z$

$Z≔z\left[1\right],z\left[2\right],z\left[3\right],z\left[4\right],z\left[5\right]$

$Z≔z_1,z_2,z_3,z_4,z_5$

$\mathrm{IncompleteBellB}\left(0\,0\,Z\right)$

$1$

$\mathrm{IncompleteBellB}\left(0\,1\,Z\right),\mathrm{IncompleteBellB}\left(1\,0\,Z\right)$

$0,0$

$\mathrm{IncompleteBellB}\left(1\,2\,Z\right),\mathrm{IncompleteBellB}\left(2\,3\,Z\right),\mathrm{IncompleteBellB}\left(3\,4\,Z\right)$

$0,0,0$

$\mathrm{IncompleteBellB}\left(n\,n\,Z\right)={Z\left[1\right]}^n$

$\mathrm{IncompleteBellB}\left(n\,n\,z_1\,z_2\,z_3\,z_4\,z_5\right)=z_1^n$

$\mathrm{eval}\left(\,n=3\right)$

$z_1^3=z_1^3$

$\mathrm{IncompleteBellB}\left(n\,k\,1\,1\,1\,1\,1\,1\,1\right)=\mathrm{Stirling2}\left(n\,k\right)$

$\mathrm{IncompleteBellB}\left(n\,k\,1\,1\,1\,1\,1\,1\,1\right)=\mathrm{Stirling2}\left(n\,k\right)$

$\mathrm{eval}\left(\,\left\{k=4\,n=7\right\}\right)$

$350=350$

$\mathrm{IncompleteBellB}\left(n\,k\,\mathrm{\%seq}\left(j!\,j=1..n-k+1\right)\right)=\mathrm{binomial}\left(n\,k\right)\mathrm{binomial}\left(n-1\,k-1\right)\left(n-k\right)!$

$\mathrm{IncompleteBellB}\left(n\,k\,\mathrm{seq}\left(j!\,j=1..n-k+1\right)\right)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)\left(n-k\right)!$

$\mathrm{eval}\left(\,\left\{k=3\,n=8\right\}\right)$

$\mathrm{IncompleteBellB}\left(8\,3\,\mathrm{seq}\left(j!\,j=1..6\right)\right)=141120$

$\mathrm{value}\left(\right)$

$141120=141120$

$\mathrm{IncompleteBellB}:-\mathrm{DiamondConvolution}\left(4\,2\,\left[Z\right]\right)$

$0,2z_1^2,6z_1{z}_2,8z_1{z}_3+6z_2^2$

$\mathrm{IncompleteBellB}\left(4\,2\,Z\right)$

$4z_1{z}_3+3z_2^2$

$\mathrm{IncompleteBellB}:-\mathrm{DiamondConvolution}\left(5\,3\,\left[Z\right]\right)$

$0,0,6z_1^3,36z_1^2{z}_2,60z_1^2{z}_3+90z_1{z}_2^2$

$\mathrm{IncompleteBellB}\left(5\,3\,Z\right)$

$10z_1^2{z}_3+15z_1{z}_2^2$

$\mathrm{CompleteBellB}\left(5\,Z\right)$

$z_1^5+10z_1^3{z}_2+10z_1^2{z}_3+15z_1{z}_2^2+5z_1{z}_4+10z_2{z}_3+z_5$

Bell, E. T. "Exponential Polynomials", Ann. Math., Vol. 35 (1934): 258-277.

The BellB, IncompleteBellB and CompleteBellB commands were introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.