Books/Resources on generating functions

Question

I'm currently doing a research on generating functions, but I have only found few books on this topic. Can anyone provide references (if possible, trying to assess the level of math competence required) on generating functions?

Ideally, what is the context of the referenced book and further details to help to put the book itself in context would be much appreciated.

Answer 1

The second edition of Herbert Wilf’s generatingfunctionology is freely available here in PDF form. Concrete Mathematics, by Graham, Knuth, and Patashnik, has a very nice introduction in Chapter $7$ and is a very fine book all around.

Answer 2

Concrete Mathematics by Knuth, Graham, and Patashnik.

Answer 3

Look into generatingfunctionology by Herbert S. Wilf.

It's precisely about generating functions! Fairly short but thorough. From the link (Wilf's website), you can

• download the $2$nd edition of the text (as a pdf document),
• download a version of the book with hyperlinks to be read on-line,
• or else click on the link to the publisher if you'd like to purchase the $3$rd edition.
• (Perhaps you can obtain a hard copy from you library.)

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From MIT (and a course taught there), you can download a pdf dedicated to generating functions, which is outlined nicely, and allows you to access the topics in order, or those of interest.

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Finally, see Ch. 10: Generating Functions of the freely available text authored by Grinstead and Laurie Snell, and published by the AMS: Introduction to Probability.

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You might want to invest in Concrete Mathematics by Graham, Knuth, and Patashnik. It has a nicely written introduction to generating functions,

Answer 4

In addition to the titles in other answers you can check Sergei Lando's book Lectures on Generating Functions

Answer 5

Another great freely (and legally!) available online resource is Flajolet and Sedgewick's Analytic Combinatorics, which in addition to having lots of material about generating functions also has lots of material about asymptotic analysis.

Answer 6

The book "A = B" which is freely downloadable at http://www.math.upenn.edu/~wilf/AeqB.html.

Answer 7

George Andrew's book The Theory of Partitions uses generating functions a lot. The book also discusses restricted and other types of partitions which require ever more interesting types of generating functions. This may help in understanding some applications of generating functions.

Answer 8

and of course the familiar ' The art of computer programming ' volume I Fundamental Algorithms by Donald Knuth beginning p87 and esp the exercises beginning p94 which in my opinion has too many thereof and realistically are not going to have time to do most of them yourself so just use his answers to those exercises beginning p496 mainly as continuation of part of his teaching tools on the subject.

Also the text by Manuel Kauers . Peter Paule The Concrete Tetrahedron about Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates SpringerWienNewYork which is quite interesting

Answer 9

Chattamvelli and Shanmugam's Generating functions in engineering and the applied sciences, published by Morgan and Claypool in 2019 is a good start for anyone interested in applications of GF.