Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, is used in the study of set partitions. It is related to Stirling and Bell numbers. It also occurs in many applications, such as in the Faà di Bruno's formula.
Bell polynomials
Exponential Bell polynomial
The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by
where the sum is taken over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that these two conditions are satisfied:
The sum
is sometimes called the nth complete exponential Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials Bn,k are sometimes called partial or incomplete Bell polynomials.
Ordinary Bell polynomial
Likewise, the partial ordinary Bell polynomial, in contrast to the usual exponential Bell polynomial defined above, is given by
where the sum runs over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that
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