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BellB

the Bell polynomials

IncompleteBellB

the incomplete Bell polynomials

CompleteBellB

the complete Bell polynomials Calling Sequence BellB($n,z$) IncompleteBellB($n,k,{z}_{1},{z}_{2},\mathrm{...},{z}_{n=k+1}$) IncompleteBellB[DiamondConvolution]($n,k,\left[{z}_{1},{z}_{2},\mathrm{...},{z}_{n-k+1}\right]$) CompleteBellB($n,{z}_{1},{z}_{2},\mathrm{...},{z}_{n}$) Parameters

 $n,k$ - non-negative integers, or algebraic expressions representing them $z,{z}_{1},\mathrm{...},{z}_{n}$ - the main variables of the polynomials, or algebraic expressions representing them Description

 • 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

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

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

$z\diamond z={\left(z\diamond z\right)}_{1},{\left(z\diamond z\right)}_{2},{\left(z\diamond z\right)}_{3},...$

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

${\left(z\diamond z\right)}_{n}=\sum _{j=1}^{n-1}\left(\genfrac{}{}{0}{}{n}{j}\right){z}_{j}{z}_{n-j}$

 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

$\mathrm{IncompleteBellB}\left(n,k,{z}_{1},{z}_{2},...\right)=\frac{{\left(z\diamond ...\diamond z\right)}_{n}}{k!}$

 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

$\mathrm{CompleteBellB}\left(n,{z}_{1},{z}_{2},\mathrm{...},{z}_{n}\right)={\sum }_{k=1}^{n}\mathrm{IncompleteBellB}\left(n,k,{z}_{1},{z}_{2},\mathrm{...},{z}_{n-k+1}\right)$

 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

$\frac{{ⅆ}^{n}}{ⅆ{x}^{n}}f\left(g\left(x\right)\right)=\sum _{k=0}^{n}{f}^{\left(k\right)}\left(g\left(x\right)\right)\mathrm{IncompleteBellB}\left(n,k,g\prime \left(x\right),g\prime \prime \left(x\right),\mathrm{...},{g}^{\left(n-k+1\right)}\left(x\right)\right)$

 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

$ⅇ{}^{\sum _{n=1}^{\mathrm{\infty }}\frac{{a}_{n}{z}^{n}}{n!}}=\sum _{n=0}^{\mathrm{\infty }}{z}^{n}\mathrm{CompleteBellB}\left(n,{a}_{1},\mathrm{...},{a}_{n}\right)$

 and in the following exponential generating function

${ⅇ}^{\left({ⅇ}^{t}-1\right)z}=\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{BellB}\left(n,z\right){t}^{n}}{n!}$ Examples

The Bell functions only evaluate to a polynomial when the arguments specifying the degree are positive integers

 > $\mathrm{BellB}\left(n,z\right)=\mathrm{Sum}\left(\mathrm{Stirling2}\left(n,k\right){z}^{k},k=0..n\right)$
 ${\mathrm{BellB}}{}\left({n}{,}{z}\right){=}{\sum }_{{k}{=}{0}}^{{n}}{}{\mathrm{Stirling2}}{}\left({n}{,}{k}\right){}{{z}}^{{k}}$ (1)
 > $\mathrm{eval}\left(,n=4\right)$
 ${{z}}^{{4}}{+}{6}{}{{z}}^{{3}}{+}{7}{}{{z}}^{{2}}{+}{z}{=}{\sum }_{{k}{=}{0}}^{{4}}{}{\mathrm{Stirling2}}{}\left({4}{,}{k}\right){}{{z}}^{{k}}$ (2)
 > $\mathrm{value}\left(\right)$
 ${{z}}^{{4}}{+}{6}{}{{z}}^{{3}}{+}{7}{}{{z}}^{{2}}{+}{z}{=}{{z}}^{{4}}{+}{6}{}{{z}}^{{3}}{+}{7}{}{{z}}^{{2}}{+}{z}$ (3)

A sequence with the values of $\mathrm{BellB}\left(n,z\right)$ for $n=0..3$

 > $\mathrm{seq}\left('\mathrm{BellB}'\left(n,z\right)=\mathrm{BellB}\left(n,z\right),n=0..3\right)$
 ${\mathrm{BellB}}{}\left({0}{,}{z}\right){=}{1}{,}{\mathrm{BellB}}{}\left({1}{,}{z}\right){=}{z}{,}{\mathrm{BellB}}{}\left({2}{,}{z}\right){=}{{z}}^{{2}}{+}{z}{,}{\mathrm{BellB}}{}\left({3}{,}{z}\right){=}{{z}}^{{3}}{+}{3}{}{{z}}^{{2}}{+}{z}$ (4)

The IncompleteBellB polynomials have a special form for some particular values of the function's parameters. For illustration purposes consider the generic sequence

 > $Z≔z\left[1\right],z\left[2\right],z\left[3\right],z\left[4\right],z\left[5\right]$
 ${Z}{≔}{{z}}_{{1}}{,}{{z}}_{{2}}{,}{{z}}_{{3}}{,}{{z}}_{{4}}{,}{{z}}_{{5}}$ (5)
 > $\mathrm{IncompleteBellB}\left(0,0,Z\right)$
 ${1}$ (6)

For $n=0$ and $0, or $0 and $k=0$, or $n, IncompleteBellB is equal to 0

 > $\mathrm{IncompleteBellB}\left(0,1,Z\right),\mathrm{IncompleteBellB}\left(1,0,Z\right)$
 ${0}{,}{0}$ (7)
 > $\mathrm{IncompleteBellB}\left(1,2,Z\right),\mathrm{IncompleteBellB}\left(2,3,Z\right),\mathrm{IncompleteBellB}\left(3,4,Z\right)$
 ${0}{,}{0}{,}{0}$ (8)

For $n=k$, the following identity holds

 > $\mathrm{IncompleteBellB}\left(n,n,Z\right)={Z\left[1\right]}^{n}$
 ${\mathrm{IncompleteBellB}}{}\left({n}{,}{n}{,}{{z}}_{{1}}{,}{{z}}_{{2}}{,}{{z}}_{{3}}{,}{{z}}_{{4}}{,}{{z}}_{{5}}\right){=}{{z}}_{{1}}^{{n}}$ (9)
 > $\mathrm{eval}\left(,n=3\right)$
 ${{z}}_{{1}}^{{3}}{=}{{z}}_{{1}}^{{3}}$ (10)

If ${z}_{j}=1$ for all $j$, the following identity holds

 > $\mathrm{IncompleteBellB}\left(n,k,1,1,1,1,1,1,1\right)=\mathrm{Stirling2}\left(n,k\right)$
 ${\mathrm{IncompleteBellB}}{}\left({n}{,}{k}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}\right){=}{\mathrm{Stirling2}}{}\left({n}{,}{k}\right)$ (11)
 > $\mathrm{eval}\left(,\left\{k=4,n=7\right\}\right)$
 ${350}{=}{350}$ (12)

If ${z}_{j}=j!$ for all $j=1,\mathrm{..},n-k+1$, the following identity, here expressed in terms of the inert sequence %seq, holds

 > $\mathrm{IncompleteBellB}\left(n,k,\mathrm{%seq}\left(j!,j=1..n-k+1\right)\right)=\mathrm{binomial}\left(n,k\right)\mathrm{binomial}\left(n-1,k-1\right)\left(n-k\right)!$
 ${\mathrm{IncompleteBellB}}{}\left({n}{,}{k}{,}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%seq}}{}\left({\mathrm{factorial}}{}\left({j}\right){,}{j}{=}{1}{..}{n}{-}{k}{+}{1}\right)\right]\right)\right){=}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{n}{-}{1}}{{k}{-}{1}}\right){}\left({n}{-}{k}\right){!}$ (13)
 > $\mathrm{eval}\left(,\left\{k=3,n=8\right\}\right)$
 ${\mathrm{IncompleteBellB}}{}\left({8}{,}{3}{,}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%seq}}{}\left({\mathrm{factorial}}{}\left({j}\right){,}{j}{=}{1}{..}{6}\right)\right]\right)\right){=}{141120}$ (14)
 > $\mathrm{value}\left(\right)$
 ${141120}{=}{141120}$ (15)

The diamond operation that enters the definition of IncompleteBellB can be invoked directly as IncompleteBellB:-DiamondConvolution. These are the first 4 elements of $z\diamond z$, a diamond operation involving 2 factors

 > $\mathrm{IncompleteBellB}:-\mathrm{DiamondConvolution}\left(4,2,\left[Z\right]\right)$
 ${0}{,}{2}{}{{z}}_{{1}}^{{2}}{,}{6}{}{{z}}_{{1}}{}{{z}}_{{2}}{,}{8}{}{{z}}_{{1}}{}{{z}}_{{3}}{+}{6}{}{{z}}_{{2}}^{{2}}$ (16)

Note that when calling IncompleteBellB:-DiamondConvolution, you pass the sequence $Z$ enclosed in a list. The value of $\mathrm{IncompleteBellB}\left(4,2,Z\right)$ is equal to the 4th element of the above sequence divided by $2!$

 > $\mathrm{IncompleteBellB}\left(4,2,Z\right)$
 ${4}{}{{z}}_{{1}}{}{{z}}_{{3}}{+}{3}{}{{z}}_{{2}}^{{2}}$ (17)

These are the first 5 elements of $z\diamond z\diamond z$, a diamond operation involving 3 factors and the value of $\mathrm{IncompleteBellB}\left(5,3,Z\right)$

 > $\mathrm{IncompleteBellB}:-\mathrm{DiamondConvolution}\left(5,3,\left[Z\right]\right)$
 ${0}{,}{0}{,}{6}{}{{z}}_{{1}}^{{3}}{,}{36}{}{{z}}_{{1}}^{{2}}{}{{z}}_{{2}}{,}{60}{}{{z}}_{{1}}^{{2}}{}{{z}}_{{3}}{+}{90}{}{{z}}_{{1}}{}{{z}}_{{2}}^{{2}}$ (18)
 > $\mathrm{IncompleteBellB}\left(5,3,Z\right)$
 ${10}{}{{z}}_{{1}}^{{2}}{}{{z}}_{{3}}{+}{15}{}{{z}}_{{1}}{}{{z}}_{{2}}^{{2}}$ (19)

The value of $\mathrm{CompleteBellB}\left(5,Z\right)$ is obtained by adding the values of $\mathrm{IncompleteBellB}\left(5,k,Z\right)$ for $k=1..5$ as explained in the Description

 > $\mathrm{CompleteBellB}\left(5,Z\right)$
 ${{z}}_{{1}}^{{5}}{+}{10}{}{{z}}_{{1}}^{{3}}{}{{z}}_{{2}}{+}{10}{}{{z}}_{{1}}^{{2}}{}{{z}}_{{3}}{+}{15}{}{{z}}_{{1}}{}{{z}}_{{2}}^{{2}}{+}{5}{}{{z}}_{{1}}{}{{z}}_{{4}}{+}{10}{}{{z}}_{{2}}{}{{z}}_{{3}}{+}{{z}}_{{5}}$ (20) References

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

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