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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.

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generatingfunctionology – anon Jan 7 at 1:06
Be also aware that generating function is very related to the Z-transform (much used in discrete signal processing) – leonbloy Jan 7 at 1:18
More advanced: "Combinatorial Species and Tree-like Structures" by Bergeron, Labelle, Leroux – Michael Greinecker Jan 7 at 7:52

8 Answers

up vote 17 down vote accepted

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.

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up vote 11 down vote

Concrete Mathematics by Knuth, Graham, and Patashnik.

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up vote 11 down vote

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.)

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.

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.

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

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up vote 9 down vote

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.

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It’s at a higher level than Wilf’s book, which in turn is at a higher level than Concrete Mathematics, but I second this recommendation. – Brian M. Scott Jan 7 at 1:22
up vote 2 down vote

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

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

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the book cannot be downloaded? – Idonknow Jan 7 at 3:31
Downloading worked for me. – MJD Jan 7 at 4:26
up vote 1 down vote

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.

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up vote 0 down vote

yes Refer the quantitative aptitude by R.S.Agrawal

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-1. Seems off-topic. – Did Jan 7 at 7:50

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