- Bell polynomials
the sum extending over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that
- 1 Convolution identity
- 2 Complete Bell polynomials
- 3 Combinatorial meaning
- 4 Properties
- 5 Applications of Bell polynomials
- 6 Software
- 7 See also
- 8 References
For sequences xn, yn, n = 1, 2, ..., define a sort of convolution by:
Note that the bounds of summation are 1 and n − 1, not 0 and n .
Let be the nth term of the sequence
For example, let us compute B4,3(x1,x2). We have
Complete Bell polynomials
is sometimes called the nth complete Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials Bn, k defined above are sometimes called "partial" Bell polynomials. The complete Bell polynomials satisfy the following identity
If the integer n is partitioned into a sum in which "1" appears j1 times, "2" appears j2 times, and so on, then the number of partitions of a set of size n that collapse to that partition of the integer n when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
For example, we have
because there are
- 6 ways to partition a set of 6 as 5 + 1,
- 15 ways to partition a set of 6 as 4 + 2, and
- 10 ways to partition a set of 6 as 3 + 3.
because there are
- 15 ways to partition a set of 6 as 4 + 1 + 1,
- 60 ways to partition a set of 6 as 3 + 2 + 1, and
- 15 ways to partition a set of 6 as 2 + 2 + 2.
Stirling numbers and Bell numbers
The value of the Bell polynomial Bn,k(x1,x2,...) when all xs are equal to 1 is a Stirling number of the second kind:
is the nth Bell number, which is the number of partitions of a set of size n.
Applications of Bell polynomials
Faà di Bruno's formula
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
The complete Bell polynomials appear in the exponential of a formal power series:
See also exponential formula.
Moments and cumulants
Representation of polynomial sequences of binomial type
For any sequence a1, a2, a3, ... of scalars, let
Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity
for n ≥ 0. In fact we have this result:
- Theorem: All polynomial sequences of binomial type are of this form.
If we let
taking this power series to be purely formal, then for all n,
- Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in Mathematica as BellY.
- Bell matrix
- Eric Temple Bell (1927–1928). "Partition Polynomials". Annals of Mathematics 29 (1/4): 38–46. doi:10.2307/1967979. JSTOR 1967979. MR1502817.
- Louis Comtet (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht, Holland / Boston, U.S.: Reidel Publishing Company.
- Steven Roman. The Umbral Calculus. Dover Publications.
- Khristo N. Boyadzhiev (2009). "Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals". Abstract and Applied Analysis 2009: Article ID 168672. doi:10.1155/2009/168672. http://www.hindawi.com/journals/aaa/2009/168672.html. (contains also elementary review of the concept Bell-polynomials)
- Silvia Noschese, Paolo E. Ricci (2003). "Differentiation of Multivariable Composite Functions and Bell Polynomials". Journal of Computational Analysis and Applications 5 (3): 333–340. doi:10.1023/A:1023227705558. http://www.springerlink.com/content/m815387gx128r1w4/. '
- Vassily G.Voinov, Mikhail S.Nikulin (1994). "On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications". Kybernetika 30 (3): 343–358. ISSN 5954 0023 5954.
- Kruchinin, V.V., 2011 , Derivation of Bell Polynomials of the Second Kind(ArXiv)
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