Contents

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

[itex]{d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1!\,m_2!\,m_3!\,\cdots 1!^{m_1}\,2!^{m_2}\,3!^{m_3}\,\cdots} f^{(m_1+\cdots+m_n)}(g(x)) \prod_{j\,:\,m_j\neq 0}g^{(m_j)}(x),[itex]

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

[itex]1m_1+2m_2+3m_3+\cdots+nm_n=n.\,[itex]

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:

[itex]{d^n \over dx^n} f(g(x))

=\sum \frac{n!}{1!^{m_1}\,2!^{m_2}\,3!^{m_3}\,\cdots} f^{(m_1+\cdots+m_n)}(g(x)) \prod_{j\,:\,m_j\neq 0}\left(g^{(m_j)}(x)/m_j!\right).[itex]

Combinatorial form

The formula has a "combinatorial" form:

[itex]{d^n \over dx^n} f(g(x))=(f\circ g)^{(n)}(x)=\sum_{\pi\in\Pi} f^{(\left|\pi\right|)}(g(x))\cdot\prod_{B\in\pi}g^{(\left|B\right|)}(x)[itex]

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).

Explication via an example

The combinatorial form may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:

[itex]\;(f\circ g)''''(x)

= f''''(g(x))g'(x)^4 + 6f'''(g(x))g''(x)g'(x)^2 \;[itex]

[itex]

\;+\; 3f''(g(x))g''(x)^2

+   4f''(g(x))g'''(x)g'(x) \;[itex]
[itex]

\;+\; f'(g(x))g''''(x). \;[itex]

What is the pattern?

[itex]\begin{matrix}
g'(x)^4

& \leftrightarrow & 1+1+1+1 & \leftrightarrow & f''''(g(x)) & \leftrightarrow & 1 \\ \\

g''(x)g'(x)^2

& \leftrightarrow & 2+1+1 & \leftrightarrow & f'''(g(x)) & \leftrightarrow & 6 \\ \\ g''(x)^2 & \leftrightarrow & 2+2 & \leftrightarrow & f''(g(x)) & \leftrightarrow & 3 \\ \\ g'''(x)g'(x) & \leftrightarrow & 3+1 & \leftrightarrow & f''(g(x)) & \leftrightarrow & 4 \\ \\ g''''(x) & \leftrightarrow & 4 & \leftrightarrow & f'(g(x)) & \leftrightarrow & 1 \end{matrix}[itex]

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.

Formal power series version

In the formal power series

[itex]f(x)=\sum_n {a_n \over n!}x^n,[itex]

we have the nth derivative at 0:

[itex]f^{(n)}(0)=a_n. \;[itex]

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

[itex]g(x)=1+\sum_{n=1}^\infty {b_n \over n!} x^n[itex]

and

[itex]f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n[itex]

and

[itex]g(f(x))=h(x)=\sum_{n=1}^\infty{c_n \over n!}x^n,[itex]

then the coefficient cn (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

[itex]c_n=\sum_{\pi=\left\{\,B_1,\,\dots,\,B_k\,\right\}} a_{\left|B_1\right|}\cdots a_{\left|B_k\right|} b_k[itex]

where π runs through the set of all partitions of the set { 1, ..., n } and B1, ..., Bk are the blocks of the partition π, and | Bj | 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.

We can also write

[itex]g(f(x)) = \sum_{n=1}^\infty

{\sum_{k=1}^{n} b_k B_{n,k}(a_1,\dots,a_{n-k+1}) \over n!} x^n.[itex]

where the expressions

[itex]B_{n,k}(a_1,\dots,a_{n-k+1})[itex]

are Bell polynomials.

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.

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

[itex]n=\underbrace{1+\cdots+1}_{m_1\ \mbox{terms}}

+\underbrace{2+\cdots+2}_{m_2\ \mbox{terms}} [itex]

[itex]

+\; \underbrace{3+\cdots+3}_{m_3\ \mbox{terms}}+\cdots[itex]

of the integer n is equal to

[itex]\frac{n!}{m_1!\,m_2!\,m_3!\,\cdots 1!^{m_1}\,2!^{m_2}\,3!^{m_3}\,\cdots}.[itex]

These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants.

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

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