They also occur in many applications, such as in Faà di Bruno's formula.
Likewise, the partial ordinary Bell polynomial is defined by where the sum runs over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that Thanks to the first condition on indices, we can rewrite the formula as where we have used the multinomial coefficient.
The exponential Bell polynomial encodes the information related to the ways a set can be partitioned.
Similarly, x1 indicates the presence of a block with a single element.
The exponent of xij indicates that there are j such blocks of size i in a single partition.
Since any set can be divided into a single block in only one way, the above interpretation would mean that Bn,1 = xn.
Similarly, since there is only one way that a set with n elements be divided into n singletons, Bn,n = x1n.
The sum of the subscripts in a monomial is equal to the total number of elements.
Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers.
Indeed, the subscripts of the variables in a monomial are the same as those given by the integer partition, indicating the sizes of the different blocks.
The total number of monomials appearing in a complete Bell polynomial Bn is thus equal to the total number of integer partitions of n. Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into.
Thus, given a complete Bell polynomial Bn, we can separate the partial Bell polynomial Bn,k by collecting all those monomials with degree k. Finally, if we disregard the sizes of the blocks and put all xi = x, then the summation of the coefficients of the partial Bell polynomial Bn,k will give the total number of ways that a set with n elements can be partitioned into k blocks, which is the same as the Stirling numbers of the second kind.
Also, the summation of all the coefficients of the complete Bell polynomial Bn will give us the total number of ways a set with n elements can be partitioned into non-overlapping subsets, which is the same as the Bell number.
For example, we have because the ways to partition a set of 6 elements as 2 blocks are Similarly, because the ways to partition a set of 6 elements as 3 blocks are Below is a triangular array of the incomplete Bell polynomials
: The exponential partial Bell polynomials can be defined by the double series expansion of its generating function: In other words, by what amounts to the same, by the series expansion of the k-th power: The complete exponential Bell polynomial is defined by
, or in other words: Thus, the n-th complete Bell polynomial is given by Likewise, the ordinary partial Bell polynomial can be defined by the generating function Or, equivalently, by series expansion of the k-th power: See also generating function transformations for Bell polynomial generating function expansions of compositions of sequence generating functions and powers, logarithms, and exponentials of a sequence generating function.
[1] The complete Bell polynomials can be recurrently defined as with the initial value
The partial Bell polynomials can also be computed efficiently by a recurrence relation: where In addition:[2] When
can be expressed as the value of the complete Bell polynomial on all arguments being x: If we define then we have the inverse relationship
The complete Bell polynomial can be expressed as determinants: and For sequences xn, yn, n = 1, 2, ..., define a convolution by: The bounds of summation are 1 and n − 1, not 0 and n .
The first few complete Bell polynomials are: Faà di Bruno's formula may be stated in terms of Bell polynomials as follows: Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows.
Suppose Then In particular, the complete Bell polynomials appear in the exponential of a formal power series: which also represents the exponential generating function of the complete Bell polynomials on a fixed sequence of arguments
Let two functions f and g be expressed in formal power series as such that g is the compositional inverse of f defined by g(f(w)) = w or f(g(z)) = z.
If f0 = 0 and f1 ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as[8] with
Consider the integral of the form where (a,b) is a real (finite or infinite) interval, λ is a large positive parameter and the functions f and g are continuous.
Assume that as x → a+, with α > 0, Re(β) > 0; and that the expansion of f can be term wise differentiated.
Then, Laplace–Erdelyi theorem states that the asymptotic expansion of the integral I(λ) is given by where the coefficients cn are expressible in terms of an and bn using partial ordinary Bell polynomials, as given by Campbell–Froman–Walles–Wojdylo formula: The elementary symmetric polynomial
For instance, together with Cayley–Hamilton theorem they lead to expression of the determinant of a n × n square matrix A in terms of the traces of its powers: The cycle index of the symmetric group
can be expressed in terms of complete Bell polynomials as follows: The sum is the nth raw moment of a probability distribution whose first n cumulants are κ1, ..., κn.
For any sequence a1, a2, …, an of scalars, let Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity More generally, we have this result: If we define a formal power series then for all n, Bell polynomials are implemented in: