The Bell polynomials (studied extensively by E.T. Bell [a2]) arise naturally from differentiating a composite function 
times, but in this context they predate Bell since they are implicit in the work of F. Faà di Bruno [a4]. Accounts of Faà di Bruno's formula, however, often fail to mention any connection with Bell polynomials. The polynomials also occur in other places without being referred to by name; in [a1] (Table 24.2), for example, the numbers 
are coefficients of partial Bell polynomials, but are not identified as such.
Suppose that 
and let
then by repeated application of the chain rule:
In general,
 | (a1) |
where 
is a homogeneous polynomial of degree 
and weight 
in the 
, known as a (partial) Bell polynomial (see [a3] for a table for 
); it has integral coefficients. Because of the homogeneity, for fixed 
all 
(
) can be determined uniquely even if the 
are omitted. Hence the (complete) Bell polynomial 
is usually defined for 
by
In [a7], however, the term Bell polynomial is used for
so the 
are included in the definition.
The following definitions are also made: 
, 
(
), 
.
Although the 
were introduced as derivatives, the Bell polynomials themselves, considered purely as polynomials in the variables 
are independent of the initial functions 
and 
. Hence information can be deduced from special choices such as 
, which gives
An alternative approach which gives the same polynomials 
is adopted in [a3], where they are defined as coefficients in the expansion of the two-variable generating function
This approach obviates the earlier assumption that the 
are derivatives.
The generating function for the complete polynomials is
Explicit formulas are known for Bell polynomials and they are examples of partition polynomials (multivariable polynomials which can be expressed as a sum of monomials, where the sum is over a set of partitions of 
; cf. also Partition). The partial polynomial
where the sum is over all partitions of 
into exactly 
non-negative parts, i.e., over all solutions in non-negative integers 
of the two equations
Since, for each fixed 
, there can be no parts of size greater than 
, the formula is often stated in the simpler looking, but equivalent, form (where necessarily 
):
 | (a2) |
where the sum is over all solutions in non-negative integers of the equations
The complete polynomial
where the sum is over all partitions of 
into arbitrarily many non-negative parts, i.e., over all non-negative integer solutions of the single equation
There are many recurrence relations for Bell polynomials, as well as formulas connecting them with other special polynomials and numbers; the following is a small selection, and others may be found in [a3], [a7], [a8].
where 
and 
are Stirling numbers of the first and second kinds (cf. Combinatorial analysis), respectively;
where 
is the Lah number;
where 
are the Bell numbers.
Combining equations (a1) and (a2) gives Faà di Bruno's formula for the 
th derivative of a composite function:
summed over all solutions in non-negative integers of
(For a generalization to functions of several variables, see [a5].)
The formula can be used, in particular, to express functions of power series as power series. If
and
then
and Faà di Bruno's formula can be used to find 
. For example, if 
, then
Hence, applying the formula and evaluating the result at 
gives
 | (a3) |
Thus, provided that 
,
A special case of (a3) is used [a6] to express the cumulants (semi-invariants, cf. Semi-invariant) 
of a probability distribution 
in terms of its moments (cf. Moment)
The probability generating function of the distribution is
and it is easy to show that
The cumulants 
(
) and their exponential generating function 
are defined in terms of 
by
Since 
, it follows from (a3) that
Similarly, starting from
and applying Faà di Bruno's formula with 
and 
(and noting that in this case 
for all 
), the inverse relation expressing moments in terms of cumulants reduces to
References
| [a1] | M. Abramowitz, I. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) |
| [a2] | E.T. Bell, "Exponential polynomials" Ann. of Math. , 35 (1934) pp. 258–277 |
| [a3] | L. Comtet, "Advanced combinatorics" , Reidel (1974) |
| [a4] | F. Faà di Bruno, "Note sur une nouvelle formule du calcul différentiel" Quart. J. Math. , 1 (1855) pp. 359–360 |
| [a5] | I.J. Good, "The multivariate saddlepoint method and chi-squared for the multinomial distribution" Ann. Math. Stat. , 32 (1961) pp. 535–548 |
| [a6] | E. Lukács, "Applications of Faà di Bruno's formula in mathematical statistics" Amer. Math. Monthly , 62 (1955) pp. 340–348 |
| [a7] | J. Riordan, "Combinatorial identities" , Wiley (1968) |
| [a8] | S. Roman, "The umbral calculus" , Acad. Press (1984) |
