complete Bell polynomials

URI: https://gptkb.org/entity/complete_Bell_polynomials
GPTKB entity
Statements (27)
Predicate Object
gptkbp:instanceOf gptkb:mathematical_concept
gptkb:polynomial_sequence
gptkbp:application differentiation of composite functions
moments and cumulants in probability theory
gptkbp:defines sum over all partitions of n of products of x_k
gptkbp:field gptkb:combinatorics
mathematical analysis
gptkbp:first_terms B_0 = 1
gptkbp:generatingFunction exp(sum_{k=1}^∞ x_k t^k / k!)
gptkbp:hasSpecialCase Bell numbers when x_k = 1 for all k
gptkbp:namedAfter gptkb:Eric_Temple_Bell
gptkbp:notation B_n(x_1, x_2, ..., x_n)
gptkbp:OEIS A099594
gptkbp:recurrence B_{n+1}(x_1,...,x_{n+1}) = sum_{k=0}^n (n choose k) B_{n-k}(x_1,...,x_{n-k}) x_{k+1}
gptkbp:relatedTo gptkb:Bell_numbers
partial Bell polynomials
gptkbp:seeAlso gptkb:exponential_polynomials
gptkb:Stirling_numbers_of_the_second_kind
gptkbp:usedFor gptkb:Faà_di_Bruno's_formula
partitioning sets
gptkbp:variant n
x_1
x_2
x_k
gptkbp:bfsParent gptkb:Bell_polynomials
gptkbp:bfsLayer 6
https://www.w3.org/2000/01/rdf-schema#label complete Bell polynomials