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, and was beatified by the Pope a century after his death. Perhaps the most well-known form a Faà di Bruno's formula says that History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... Francesco Faà di Bruno (1825—1888) was an Italian mathematician and priest, born at Alessandria. ... In Catholicism, beatification (from Latin beatus, blessed, via Greek μακαριος, makarios) is a recognition accorded by the church of a dead persons accession to Heaven and capacity to intercede on behalf of individuals who pray in their name (intercession of saints). ...

where the sum is over all n-tuples (m₁, ..., m_n) satisfying the constraint

Sometimes, to give it a pleasing and memorable pattern, it is written in a way that is slightly less explicit about the values of the coefficients of the various derivatives:

Combinatorial form

The formula has a "combinatorial" form:

where

• π runs through 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).

A partition of U into 6 blocks: a Venn diagram representation. ...

Explication via an example

The combinatorial form 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. In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ... A partition of U into 6 blocks: a Venn diagram representation. ...

Similarly for the other terms. That is the pattern.

Formal power series version

In the formal power series In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

regardless of the bounds of summation (i.e. regardless of whether k runs from 0 to ∞ or from 1 to ∞, etc.), we have the nth derivative at 0:

This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context.

If

and

and

then the coefficient c_n (which would be the nth derivative of h evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by

where π runs through the set of all partitions of the set { 1, ..., n } and B₁, ..., B_k are the blocks of the partition π, and | B_j | is the number of members of the jth block, for j = 1, ..., k.

This version of the formula is particularly well suited to the purposes of combinatorics. See the "compositional formula" in Chapter 5 of Enumerative Combinatorics, Volumes 1 and 2 (http://www-math.mit.edu/~rstan/ec/), Richard P. Stanley, Cambridge University Press, 1997 and 1999, ISBN 0-521-55309-1N. Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ...

A special case

If f(x) = e^x 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. Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ... See also moment (physics). ...

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 integer partition A partition of U into 6 blocks: a Venn diagram representation. ... In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...

of the integer n is equal to

These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants. In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are given by the sum extending over all sequences j1, j2, j3, ..., jn−k+1 of positive integers such that Combinatorial meaning If the integer n is partitioned into a sum in which 1 appears j1 times... Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ...

External links

The Curious History of Faà di Bruno's Formula (http://www.maa.org/news/monthly217-234.pdf)