Complex analysis and generating functions problem or question

Question

I've been studying some complex analysis lately and I find very intriguing the relation between this field and generating functions. I've seen this video by 3Blue1Brown and it's really fascinating. Essentially, I'd like to find some kind of problem or question in which I could use this types of techniques in order to ilustrate the procedures for a project. I'd like to remark that this is a project, not actual research, so I just need to find a problem so I can apply this techniques and show some further applications. Ideally, the problem is cloesly related to something like computer science, or anything that allows me to create something, since the project is meant to be original and I'm not so naive to belive that with my knowledge I could discover something new.

Answer 1

Here is one of the simplest examples I know where one really benefits from using complex analysis (as opposed to just the complex numbers + formal power series, which already has various nice applications). Let $a_n$ be the number of ways $n$ horses can place in a race with ties; equivalently, this counts the number of weak orderings on a set with $n$ elements. This sequence begins

$$1, 1, 3, 13, 75, 541, \dots $$

and is A000670 on the OEIS, which calls them the Fubini numbers; they are also called the ordered Bell numbers. General concepts in the theory of generating functions can be used to write down a generating function here: a weak ordering is a "sequence of sets," which gives an exponential generating function

$$A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} = \frac{1}{2 - e^z}.$$

Now given this generating function, we can apply what Flajolet and Sedgewick call "meromorphic singularity analysis": we consider $A(z)$ as a meromorphic function and ask for its dominant pole. The poles occur at $z = \log 2 + 2 \pi i k, k \in \mathbb{Z}$, they are all simple, and the dominant one occurs at $k = 0$, with residue

$$\lim_{z \to \log 2} \frac{z - \log 2}{2 - e^z} = - \frac{1}{2}$$

which gives

$$A(z) = \frac{1}{2 \log 2 \left( 1 - \frac{z}{\log 2} \right)} + \text{other poles}.$$

Expanding this out gives the dominant asymptotic

$$\boxed{ a_n \approx \frac{n!}{2 (\log 2)^{n+1}}. }$$

Imagine trying to prove this directly from the definition of a weak ordering! Moreover, the generating function tells us that the error term of this approximation is quasi-periodic; it is determined by the next most dominant poles at $\log 2 \pm 2 \pi i$, which are complex. This is really inaccessible directly from the combinatorics (as far as I know, anyway). I should also say that this approximation is extremely accurate already for small values of $n$; for example, for $n = 5$ it gives $a_5 \approx 541.00\color{red}{15} \dots$.

The most comprehensive reference for this stuff I know is Flajolet and Sedgewick's Analytic Combinatorics; if it has one downside it can be a little overwhelming at first glance because there is just so much material in it. This example is discussed in II.3 on page 109 and again in IV.3 on page 244.

Answer 2

As already recommended at the end of this great video presentation you might want to have a look at generatingfunctionology by H. S. Wilf.

• You will find there instructive examples dealing with series multisection. Take a look at (2.4.5) to (2.4.9). See for instance Example 4 where we are looking for solutions of \begin{align*} \lambda_n=\sum_{k=0}^n(-1)^k\binom{n}{3k}\qquad\qquad n\geq 0 \end{align*}

• See also Problem 25 which addresses a series multisection problem in computer science.

Note: Series multisection is also sometimes discussed at this site. See for instance this post.