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Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook,^{[1]} considered the first published reference on the subject.^{[2]}
Perhaps the most well-known form of Faà di Bruno's formula says that
where the sum is over all n-tuples of nonnegative integers (m_{1}, …, m_{n}) satisfying the constraint
Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:
Combining the terms with the same value of m_{1} + m_{2} + ... + m_{n} = k and noticing that m_{ j} has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials B_{n,k}(x_{1},...,x_{n−k+1}):
The formula has a "combinatorial" form:
where
The combinatorial form may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:
The pattern is
The factor \scriptstyle g''(x)g'(x)^2 \; corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor \scriptstyle f'''(g(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, the factor \scriptstyle g''(x)^2 \; in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), while \scriptstyle f''(g(x)) \,\! corresponds to the fact that there are two summands (2 + 2) in that partition. The coefficient 3 corresponds to the fact that there are \tfrac{1}{2}\tbinom{4}{2}=3 ways of partitioning 4 objects into groups of 2. The same concept applies to the others.
A memorizable scheme is as follows:
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
of the integer n is equal to
These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants.
Let y = g(x_{1}, ..., x_{n}). 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):^{[3]}
where (as above)
A further generalization, due to Tsoy-Wo Ma, considers the case where y is a vector-valued variable.^{[4]} The general form, for variational calculus (Gâteaux differentials are the most general form of differential), was derived in 2012.^{[5]}
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:
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.
Suppose f(x)=\sum_{n=0}^\infty {a_n} x^n and g(x)=\sum_{n=0}^\infty {b_n} x^n are formal power series and b_0 = 0.
Then the composition f \circ g is again a formal power series,
and the coefficient c_{n}, for n ≥ 1, can be expressed as a sum over compositions of n or as an equivalent sum over partitions of n :
is the set of compositions of n with k denoting the number of parts,
or
is the set of partitions of n into k parts, in frequency-of-parts form.
The first form is obtained by picking out the coefficient of x^{n} in (b_{1}x+b_{2}x^2+ \cdots)^{k} "by inspection", and the second form is then obtained by collecting like terms, or alternatively, by applying the multinomial theorem.
The special case f(x) = e^{x}, g(x) = ∑_{n ≥ 1} a_{n} /n! x^{n} gives the exponential formula. The special case f(x) = 1/(1-x), g(x) = ∑_{n ≥ 1} (-a_{n} ) x^{n} gives an expression for the reciprocal of the formal power series ∑_{n ≥ 0} a_{n} x^{n} in the case a_{0} = 1 .
Stanley ^{[6]} gives a version for exponential power series. In the formal power series
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
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 B_{1}, ..., 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.
We can also write with respect to the notation above
where B_{n,k}(a_{1},...,a_{n−k+1}) are Bell polynomials.
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.
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