2019-03-10 13:47:43 8 Comments

The intent of this question is to provide a list of learning resources for people who are new to generating functions and would like to learn about them.

I'm personally interested in combinatorics, and I sometimes use generating functions in answers to combinatorial questions on stackexchange, but I know many readers are not familiar with these objects. I hope this list will help those readers and provoke interest in GFs generally.

I will provide an answer below, but feel free to edit my answer or provide your own answer. Initially it will be a short list, but maybe it will grow over time. Please regard this question and its answers as a community resource.

Acknowledgement: In asking this self-answered question I was prompted by helpful advice provided by this discussion on mathematics meta stackexchange, users quid and John Omielan in particular. Thank you.

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## 4 comments

## @awkward 2019-03-10 13:47:43

Here are some resources to get you started on generating functions. With one exception, which is clearly designated, any of the items mentioned here should be suitable to provide a gentle introduction to GFs for total newbies.

generatingfunctionologyby Herbert S. Wilf is probably the best introductory text. You can find this book in pdf format for free, online, but I think it's worth adding a hard copy to your library. The writing style is breezy and entertaining. First sentence: "A generating function is a clothesline on which we hang up a sequence of numbers for display."An Introduction to the Analysis of Algorithmsby Robert Sedgewick and Philippe Flajolet is another fine introduction. Despite its title, the book is mostly about generating functions. Coursera has a free online course from Princeton University based on this book, with Professor Sedgewick as the instructor,so here is your chance to sit at the feet of the Master, figuratively: Coursera Analysis of Algorithms. There is also a follow-on course on Analytic Combinatorics: Coursera Analytic Combinatorics. The analytic combinatorics course is based on the bookAnalytic Combinatoricsby the same two authors, which is a fine book, but I don't think it's the best starting point for most beginners. (Of course, you might be the exception, and it really is a wonderful book with encyclopedic coverage.) Both Coursera courses are selective in their coverage and do not attempt to cover the entire contents of their respective books, especially the course on analytic combinatorics.While we are on the subject of free online materials, Bogart's Introductory Combinatorics (pdf) includes an introduction to generating functions. Bogart leads the reader to discover ideas and methods for himself through a series of problems. In fact, the book is almost entirely composed of problems.

Schaum's Outline of Theory and Problems of Combinatoricsby V.K. Balakrishnan includes a good introduction to generating functions and is noteworthy for being inexpensive by comparison with other texts (around \$20 on Amazon the last time I checked). I find this book useful both for reference and as a learning resource. Coverage includes some relatively advanced topics, such as rook polynomials and the Polya Enumeration Theorem, but you can skip those parts on a first reading.Applied Combinatoricsby Alan Tucker.As for prerequisites, many applications of GFs require only a knowledge of polynomials. But many others require infinite series, so you need some exposure to series like infinite geometric series, the series for $e^x$, and the Binomial Theorem for negative and fractional exponents. Interestingly enough, we can often (but not always) ignore questions of convergence, because we view the series as formal objects and don't worry about evaluating them. It is frequently useful to differentiate or integrate a generating function, so you need calculus skills. (In fact, part of the fascination of generating functions is that they take a problem in discrete mathematics, where the normal tools are addition, multiplication, subtraction and division, and transform the problem into the realm of the continuous, where tools like differential and integral calculus apply.) Some applications do use differential equations or complex analysis, but you can go a long way without these.

A computer algebra system, such as Wolfram Alpha, though not essential, is sometimes useful to take the drudgery out of calculations that would otherwise be tedious. I used to feel guilty when I used a computer to multiply two long polynomials, but I've gotten over the guilt and now feel that to the combinatorialist, computer algebra is a basic tool like a calculator.

To pique your interest in GFs, here is how the statistician Frederick Mosteller described his initial exposure to generating functions.

## @clathratus 2019-03-11 16:00:35

One book which is worth mentioning is

Irresistible Integralsby George Boros and Victor Moll. It touches a bit on GFs, especially in relation to their use in the evaluation of series and integrals, as well as their connections to the study of polynomials and recurrence relations.One of the first chapters uses the recursive definition of the Fibonacci numbers in order to find their generating function, namely $$\sum_{n\geq0}F_nx^n=\frac1{1-x-x^2}$$ But the use of GF's is consistent throughout the book. I highly recommend it if you also want to learn about series, integrals, and polynomials.

## @darij grinberg 2019-03-11 15:54:06

Nicholas Loehr's

Bijective Combinatorics, 1st edition includes the best treatment of formal power series I have ever seen in the combinatorics literature. (It has been watered down in the 2nd edition, which has recently appeared under the nameCombinatorics.)Herbert Wilf's

generatingfunctionologygoes farther than any other text I know into the usage of generating functions (but is sloppier at setting the stage).A lot of other texts focus on uses of generating functions without formally defining them. For example: Aigner or Wagner or Hulpke or MacGillivray. For best effects, combine them with some text on abstract algebra.

## @Markus Scheuer 2019-03-10 19:54:45

## @awkward 2019-03-11 12:10:29

I agree entirely, the book is a treasure. I would just like to add that if the reader is impatient to learn about GFs, it is possible to skip directly to the material on GFs without absorbing all the information in the early chapters. (I recall reading someone's complaint that he was directed to CM to learn about GFs, but he bogged down somewhere in the earlier chapters, which do contain a lot of material. Not that any of the book should be neglected, it's all valuable.)

## @Markus Scheuer 2019-03-11 12:23:07

@awkward: I agree in that

allof the book is highly informative and a great pleasure to read. :-) I think I've just added the hint you have in mind by indicating section 5.4 asstartingpoint. Section 5.4 provides some basic material which is helpful to grasp chapter 7 afterwards.