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:
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
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 ):
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 Faà di Bruno's formula can be used to find . For example, if , then
Hence, applying the formula and evaluating the result at gives
Thus, provided that ,
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
|[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)|
Bell polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell_polynomial&oldid=17635