Editing Bell polynomials (section)

==Definitions==

===Exponential Bell polynomials===
The ''partial'' or ''incomplete'' exponential Bell polynomials are a [[triangular array]] of polynomials given by

:<math>\begin{align}
B_{n,k}(x_1,x_2,\dots,x_{n-k+1}) &= \sum{n! \over j_1!j_2!\cdots j_{n-k+1}!}
\left({x_1\over 1!}\right)^{j_1}\left({x_2\over 2!}\right)^{j_2}\cdots\left({x_{n-k+1} \over (n-k+1)!}\right)^{j_{n-k+1}} \\
&= n! \sum \prod_{i=1}^{n-k+1} \frac{x_i^{j_i}}{(i!)^{j_i} j_i!},
\end{align}</math>

where the sum is taken over all sequences ''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>, ..., ''j''<sub>''n''−''k''+1</sub> of non-negative integers such that these two conditions are satisfied:

:<math>j_1 + j_2 + \cdots + j_{n-k+1} = k, </math> 
:<math>j_1 + 2 j_2 + 3 j_3 + \cdots + (n-k+1)j_{n-k+1} = n.</math>

The sum

:<math>\begin{align}
B_n(x_1,\dots,x_n)&=\sum_{k=1}^n B_{n,k}(x_1,x_2,\dots,x_{n-k+1})\\
&=n! \sum_{1j_1 +\ldots+ nj_n=n} \prod_{i=1}^n \frac{x_i^{j_i}}{(i!)^{j_i}j_i!}
\end{align}</math>

is called the ''n''th ''complete exponential Bell polynomial''.

===Ordinary Bell polynomials===
Likewise, the partial ''ordinary'' Bell polynomial is defined by
:<math>\hat{B}_{n,k}(x_1,x_2,\ldots,x_{n-k+1}) = \sum \frac{k!}{j_1! j_2! \cdots j_{n-k+1}!} x_1^{j_1} x_2^{j_2} \cdots x_{n-k+1}^{j_{n-k+1}}, </math>
where the sum runs over all sequences ''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>, ..., ''j''<sub>''n''−''k''+1</sub> of non-negative integers such that
:<math>j_1 + j_2 + \cdots + j_{n-k+1} = k,</math>
:<math>j_1 + 2 j_2 + \cdots + (n-k+1)j_{n-k+1} = n.</math>

Thanks to the first condition on indices, we can rewrite the formula as
:<math>\hat{B}_{n,k}(x_1,x_2,\ldots,x_{n-k+1}) = \sum \binom{k}{j_1, j_2, \ldots, j_{n-k+1}} x_1^{j_1} x_2^{j_2} \cdots x_{n-k+1}^{j_{n-k+1}}, </math>

where we have used the [[Multinomial theorem|multinomial coefficient]].

The ordinary Bell polynomials can be expressed in the terms of exponential Bell polynomials:
:<math>\hat{B}_{n,k}(x_1,x_2,\ldots,x_{n-k+1}) = \frac{k!}{n!}B_{n,k}(1!\cdot x_1,2!\cdot x_2,\ldots,(n-k+1)!\cdot x_{n-k+1}).</math>

In general, Bell polynomial refers to the exponential Bell polynomial, unless otherwise explicitly stated.