Bell polynomials

In the mathematical subfield of combinatorics the Bell polynomials, named after Eric Temple Bell call, the following triangular array of polynomials

Wherein the sum of all sequences J1, J2, J3, ..., Jn -K 1 non- negative integer values ​​is formed so that

• 3.1 Bell polynomials and Stirling numbers
• 3.2 convolution identity

• 4.1 Formula of Faa di Bruno
• 4.2 Moments and cumulants
• 4.3 Representation of polynomial sequences with binomial property

Complete Bell polynomials

The sum

Sometimes referred to as n-th full Bell polynomial. For better differentiation from the full - Bell polynomials are defined above polynomials Bn, k also sometimes referred to as " incomplete " Bell polynomials.

The complete Bell polynomials satisfy the following equation

Combinatorial importance

If the integer n is decomposed into a sum, in which the "1" j1 times occurs, the "2" j2 occurs times, etc., then corresponds to the number of possible partitions of a set of size n, such that the union of the original amount results, the respective coefficients of the Bell polynomial. is then the sum of all of the monomials degrees K.

Examples

It applies, for example,

Since it

A further example is

Since it

Properties

Bell polynomials and Stirling numbers

The value of Bell polynomial Bn, k ( x1, x2, ... ) if all xi "1" are the same, is a Stirling number of second kind

The sum

Corresponds to the nth Bell number, which describes the number of possible partitions of a set with n elements.

Convolution identity

For sequences xn, yn, n = 1, 2, ..., you can define a type of folding:

The limits of the sum of " 1" and " n-1" in place of " 0" and " n".

Be the nth term of the sequence

Then:

Folding the identity can be used to calculate individual Bell polynomials. The calculation of results with

And, accordingly,

Applications

Formula of Faa di Bruno

The formula of Faa di Bruno can be expressed using the Bell polynomials as follows:

In a similar manner, a power series version of the formula of Faa di Bruno can be set up. adopted

Then

The complete Bell polynomials appear in the exponential of a formal Potenzeihe on:

Moments and cumulants

The sum

Is the nth moment of a probability distribution whose first n cumulants κ1, ..., κn are. In other words, the nth moment of the nth complete Bell polynomial is evaluated at the n first cumulants.

Representation of polynomial sequences with binomial property

For an arbitrary ( scalar ) sequence a1, a2, a3, ... be

These polynomials satisfy the binomial property, ie

For n ≥ 0, it holds that all polynomial sequences which satisfy the binomial property are of this form.

If the power series

Formally accepted as a purely applies, we obtain for all n