Faà di Bruno's formula
The formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, 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:
where
- π runs throught the set Π of all partitions of the set { 1, ..., n },
- "B ∈ π" means the variable B runs through the list of all of the "blocks" of the partition π, and
- |A| denotes the cardinality of the set A (so that |π| is the number of blocks in the partition π and |B| is the size of the block B).
Explication via an example
This may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:
What is the pattern?
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.
The Faà di Bruno coefficients
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
of the integer n is equal to
A special case
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.
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