A complete Bell polynomial 
is a polynomial
that enumerates set partitions of an 
-element set, with each set
block of size 
assigned weight 
.
In the exponential convention used here, it is the sum of
the partial Bell polynomials over all possible
numbers of blocks,
 |
(1)
|
for a positive integer 
, with 
.
Ordinary normalizations have different factorial weights.
In particular, the ordinary partial family rescales both the arguments and the coefficients
of the partial Bell polynomials (Di Nardo
and Guarino 2022).
The exponential generating function is
 |
(2)
|
The first few complete Bell polynomials are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Suppressing the arguments, a recurrence relation is
 |
(7)
|
where 
is a nonnegative integer. Setting every 
gives the univariate Bell polynomial 
, while setting every 
gives the Bell number.
Complete Bell polynomials express derivatives of an exponential function through Faà di Bruno's formula,
 |
(8)
|
where 
is a positive integer and the necessary derivatives
exist. If 
is analytic and nonzero near a point,
with logarithmic derivative 
, this gives 
. Applying the
identity to 
gives formulas for complex residues at poles
of order 
(Liu 2001).
Complete Bell polynomials also express raw moments in terms of cumulants. For a random variable whose moment-generating function is defined near 0,
 |
(9)
|
where 
is a positive integer, 
is its 
th raw moment, and 
its 
th cumulant. The identity follows
from 
, where 
is the moment-generating
function and 
the cumulant-generating
function (Di Nardo and Guarino 2022).
The complete Bell polynomial is also a special case of the generalized Bell polynomial for function composition. In the latter notation,
 |
(10)
|
This follows by applying the Faà di Bruno formula to 
,
whose outer derivatives, after the common factor 
is removed, all equal 1.
