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2 Explication via an example 3 The Faà di Bruno coefficients 4 A special case |
Faà di Bruno's formula is an identity in mathematics named in honor of Francesco Faà di Bruno (1825 - 1888), who was (in chronological order) a military officer, a mathematician, and a priest. It can be stated in a general and perhaps initially forbidding form thus:
This may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:
The factor g ′ ′ (x) g ′ (x)2 corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor f ′ ′ ′ (x) that goes with it corresponds to the fact that there are three summands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.
Similarly for the other terms. That is the pattern.
These partition-counting Faà di Bruno coefficients have a "closed-form" expression. The number of partitions of a set of size n corresponding to the partition
If f(x) = ex then all of the derivatives of f are the same, and are a factor common to every term. In case g(x) is a cumulant-generating function, then f(g(x))) is a moment-generating function, and the polynomial in various derivatives of g is the polynomial that expresses the moments as functions of the cumulants.The formula
where π runs throught the list of all partitions of the set { 1, ..., n } and |A| denotes the cardinality of the set A.
Explication via an example
What is the pattern?The Faà di Bruno coefficients
of the integer n
is equal toA special case
