Editing Bell polynomials (section)

===Generating function===
The exponential partial Bell polynomials can be defined by the double series expansion of its generating function:
:<math> 
\begin{align}
\Phi(t,u) &= \exp\left( u \sum_{j=1}^\infty x_j \frac{t^j}{j!} \right) = \sum_{n\geq k \geq 0} B_{n,k}(x_1,\ldots,x_{n-k+1}) \frac{t^n}{n!} u^k\\
 &= 1 + \sum_{n=1}^\infty \frac{t^n}{n!} \sum_{k=1}^n u^k B_{n,k}(x_1,\ldots,x_{n-k+1}).
\end{align}
</math>

In other words, by what amounts to the same, by the series expansion of the ''k''-th power:
:<math> \frac{1}{k!}\left( \sum_{j=1}^\infty x_j \frac{t^j}{j!} \right)^k = \sum_{n=k}^\infty B_{n,k}(x_1,\ldots,x_{n-k+1}) \frac{t^n}{n!}, \qquad k = 0, 1, 2, \ldots  </math>
 
The complete exponential Bell polynomial is defined by <math>\Phi(t,1)</math>, or in other words:
:<math> \Phi(t,1) = \exp\left( \sum_{j=1}^\infty x_j \frac{t^j}{j!} \right) = \sum_{n=0}^\infty B_n(x_1,\ldots, x_n) \frac{t^n}{n!}.</math>

Thus, the ''n''-th complete Bell polynomial is given by
:<math> B_n(x_1,\ldots, x_n) = \left. \left(\frac{\partial}{\partial t}\right)^n \exp\left( \sum_{j=1}^n x_j \frac{t^j}{j!} \right) \right|_{t=0}. </math>

Likewise, the ''ordinary'' partial Bell polynomial can be defined by the generating function
:<math> \hat{\Phi}(t,u) = \exp \left( u \sum_{j=1}^\infty x_j t^j \right) = \sum_{n\geq k\geq 0} \hat{B}_{n,k}(x_1,\ldots,x_{n-k+1}) t^n \frac{u^k}{k!}.</math>

Or, equivalently, by series expansion of the ''k''-th power:
:<math>\left(\sum_{j=1}^\infty x_j t^j\right)^k = \sum_{n=k}^\infty \hat{B}_{n,k}(x_1, \ldots, x_{n-k+1}) t^n. </math>

See also [[Generating function transformation#Powers of an OGF and composition with functions|generating function transformations]] for Bell polynomial generating function expansions of compositions of sequence [[generating functions]] and [[Exponentiation|powers]], [[logarithm]]s, and [[exponential function|exponentials]] of a sequence generating function. Each of these formulas is cited in the respective sections of Comtet.{{Sfn|Comtet|1974}}