Faà di Bruno's formula

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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 of Faà di Bruno's formula says that

{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}\left(g^{(j)}(x)\right)^{m_j},

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

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

Sometimes, to give it a pleasing and memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

{d^n \over dx^n} f(g(x)) =\sum \frac{n!}{m_1!\,m_2!\,m_3!\,\cdots} f^{(m_1+\cdots+m_n)}(g(x)) \prod_{j\,:\,m_j\neq 0}\left(g^{(j)}(x)/j!\right)^{m_j}.

Contents

[edit] Combinatorial form

The formula has a "combinatorial" form:

{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)

where

  • "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).

[edit] Explication via an example

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

\;(f\circ g)''''(x) = f''''(g(x))g'(x)^4 + 6f'''(g(x))g''(x)g'(x)^2 \;\,\!
\;\quad\quad\quad\quad+\; 3f''(g(x))g''(x)^2 + 4f''(g(x))g'''(x)g'(x) \;\,\!
\;\quad\quad\quad\quad+\; f'(g(x))g''''(x). \;\,\!

What is the pattern?

\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}

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.

[edit] Combinatorics of 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

n=\underbrace{1+\cdots+1}_{m_1\ \mbox{terms}} +\underbrace{2+\cdots+2}_{m_2\ \mbox{terms}}
+\; \underbrace{3+\cdots+3}_{m_3\ \mbox{terms}}+\cdots

of the integer n is equal to

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

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

[edit] A multivariable version

Let y = g(x1, ..., xn). Then the following identity holds regardless of whether the n variables are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables (if it seems opaque, see the very concrete example below):

{\partial^n \over \partial x_1 \cdots \partial x_n}f(y) = \sum_{\pi\in\Pi} f^{(\left|\pi\right|)}(y)\cdot\prod_{B\in\pi} {\partial^{\left|B\right|}y \over \prod_{j\in B} \partial x_j}

where (as above)

  • "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).

See Hardy, Michael, "Combinatorics of Partial Derivatives", Electronic Journal of Combinatorics, 13 (2006), #R1.

[edit] Example

The five terms in the following expression correspond in the obvious way to the five partitions of the set { 1, 2, 3 }, and in each case the order of the derivative of f is the number of parts in the partition:

{\partial^3 \over \partial x_1\, \partial x_2\, \partial x_3}f(y) = f'(y){\partial^3 y \over \partial x_1\, \partial x_2\, \partial x_3}


+ f''(y) \left( {\partial y \over \partial x_1} \cdot{\partial^2 y \over \partial x_2\, \partial x_3} +{\partial y \over \partial x_2} \cdot{\partial^2 y \over \partial x_1\, \partial x_3} + {\partial y \over \partial x_3} \cdot{\partial^2 y \over \partial x_1\, \partial x_2}\right)


+ f'''(y) {\partial y \over \partial x_1} \cdot{\partial y \over \partial x_2} \cdot{\partial y \over \partial x_3}.

If the three variables are indistinguishable from each other, then three of the five terms above are also indistinguishable from each other, and then we have the classic one-variable formula.

[edit] Formal power series version

In the formal power series

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

we have the nth derivative at 0:

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

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

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

and

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

and

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

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

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

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, Richard P. Stanley, Cambridge University Press, 1997 and 1999, ISBN 0-521-55309-1N.

We can also write

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.

where the expressions

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

are Bell polynomials.

[edit] 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.

[edit] External links

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