Why would you take the logarithmic derivative of a generating function?

Question

Today, my climbing expedition scaled Mt. Sloane to request the Oracle's Extensive Insight into Sequences. The monks there had never heard of our plight, so they inscribed our query in mystical runes on a scrip of paper and took it into a room we were not permitted to enter. The Superseeker, as they called it, eventually responded with a fresh scroll, bearing (among other, more familiar, symbols) six imposing letters: LGDEGF.

"Logarithmic Derivative Exponential Generating Function," the monks muttered in unison as I unravelled the scroll, nodding and tittering amongst themselves. But what is such a thing? They were quick to recite that it is a function $f$ such that

$$\exp\biggl(\int f(x) \,dx\biggr) = \sum_n a_n \frac{x^n}{n!}$$

for my sequence $a_n$, and that the information in the scroll pertained to this $f$, but they refused to answer any further questions.

My expedition crew was well-versed in the basic science of generating functions, ordinary power series and exponential. But why might taking the logarithmic derivative of either generating function give interesting or exciting information? Where do they occur in the wild? Most importantly, where in the literature can we learn about them?

Answer 1

Let $X$ be a real-valued random variable. Then we have $$ \operatorname E(e^{tX}) = 1 + m_1 t + m_2 \frac{t^2} 2 + m_3 \frac {t^3} 6 + m_4 \frac{t^4}{24} + \cdots $$ and $m_k = \operatorname E(X^k)$ is the $k$th moment of the probability distribution of $X.$

The $k$th central moment of the distribution is $\mu_k(X)=\operatorname E((X-m_1)^k).$ The central moment enjoys the properties of shift invariance, which means $\mu_k(X+c) = \mu_k(X)$ for constants $c,$ and homogeneity, which means $\mu_k(cX) = c^k \mu_k (X).$ But only when $k=2\text{ or }3$ does it enjoy the property of additivity, which means that if $X_1,\ldots, X_n$ are independent random variables, then $\mu_k(X_1+\cdots+X_n) = \mu_k(X_1)+\cdots+\mu_k(X_n).$

However, for each $k\ge2$ there is a $k$th-degree polynomial in the first $k$ moments that simultaneously has all three properties. It is called the $k$th cumulant. The fourth cumulant is the fourth central moment minus $3$ times the square of the second central moment. (The second and third cumulants are merely the second and third central moments.) For $k\ge2,$ the cumulants are fully characterized by this description plus the condition that the coefficient of the $k$th moment is $1.$

Theorem: The exponential generating function of the sequence of cumulants (where the $1$st cumulant is $m_1$ as defined above, so it is shift-equivariant rather than shift-invariant like the higher cumulants) is the logarithm of the exponential generating function of the moments.

Answer 2

There is more than one way to interpret the logarithmic derivative. One way is that it is a sequence transform related to sequence recursions. For example, suppose that we have the sequences $\,a_n\,$ and $\,b_n\,$ with their associated exponential generating functions

$$ A(x) = \sum_{n=0}^\infty a_n x^n/n!, \qquad B(x) = \sum_{n=0}^\infty b_n x^n/n! $$

related such that

$$ A\,'(x) = A(x) B(x). $$

This implies the recursion relation

$$ a_{n+1} = \sum_{k=0}^n {n \choose k} a_k b_{n-k} $$

which is a recursion for the sequence $\,a_n\,$ using the binomial convolution of it with the other sequence $\,b_n.\,$

Another way to write the relation between the generating functions is that

$$ B(x) = \log(A(x))'. $$

Thus, $\,B(x)\,$ is the logarithmic derivative of $\,A(x).\,$ Inverting this relation we get that

$$ A(x) = \exp\big(\!\int\! B(x)\, dx\big). $$

A simple example of this is for OEIS sequence A000085 which is the number of permutations that are involutions. One recursion is

$$ a_{n+1} = a_n + n\, a_{n-1} $$

which corresponds to $\, b_0 = b_1 = 1. \,$ Thus, the exponential generating function of the sequence is

$$ A(x) = \exp(x + x^2/2!). $$

Another simple example is for OEIS sequence A182386 which is related to derangments. One simple recursion is

$$ a_{n+1} = -(n+1)a_n + 1, $$

but more useful for our purpose is the recursion

$$ a_{n+1} = \sum_{k=1}^n {n \choose k} (-1)^k k!\,a_{n-k} $$

which implies that the exponential generating function of the sequence $\,a_n\,$ is

$$ A(x) = \exp(x)/(1+x). $$

Answer 3

For starters, let's talk about why you might want to take the logarithm of an exponential generating function. The starting point here is the exponential formula, one version of which, roughly speaking, says that if $A(x) = \sum a_n \frac{x^n}{n!}$ is the exponential generating function of structures of some kind (e.g. graphs) which have a decomposition into connected components, then $\log A(x)$ is the exponential generating function of connected structures (e.g. connected graphs). This is a powerful and general result and has many applications, in both directions (taking logs and taking exponentials). As a simple example, the EGF for the number of ways to partition a set into subsets with cardinalities lying in some $S \subseteq \mathbb{N}$ is

$$\exp \left( \sum_{n \in S} \frac{x^n}{n!} \right).$$

The exponential formula comes in a "cyclic form" where instead of thinking of $\log A(x)$ as an exponential generating function we write it in the form $\sum b_n \frac{x^n}{n}$; see this blog post for full details. This version of the exponential formula implies, for example, that the EGF for the number of permutations in $S_n$ whose cycles have cardinalities lying in some $S \subseteq \mathbb{N}$ is

$$\exp \left( \sum_{n \in S} \frac{x^n}{n} \right).$$

This version of the exponential formula is more relevant to taking logarithmic derivatives since taking the derivative of the logarithm removes the factor of $n$.

There's a lot more to say here, including a general interpretation of what it means to compose generating functions; for more see the first half of Analytic Combinatorics.

Answer 4

One thing to add on to Michael's excellent answer.

A random variable $X$ (with all cumulants) is normal iff $\kappa_n(X)=0$ for all $n\ge 3$. Furthermore, a sequence of random variables $X_k$ (with all cumulants) converges in distribution to a normal distribution iff $\kappa_n(X_k)\to 0$ for all $n\ge 3$.

This is often in practice much easier to show than to show all moments converge to respective moments of a normal distribution.