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I'm trying to understand ordinary generating functions. I've been looking for any tutorial or some explanations about the topic but I haven't found anything useful and - what's more important - well explained.

1) Do ALL ordinary generating functions look like the polynomial 1 + a1x + a2x^2 + a3x^3 + ..., where (a)n depend on the given sequence? So for example the generating function for the (1, 0, 1, 0, ...) sequence would be 1 + a1x + a3x^3 + a5x^5 + ... ?

2) What actually is the generating function? What's the point of adding increasing powers of x to the sequence elements and summing them up?

Are there any simple excercises about this topic?

Tomorrow I have an exam, and I'm not able to understand anything about this, even though I've read a couple of websites I've found about generating functions.

Sorry if I've mistaken some math words, I haven't been taught math in english.

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2 Answers 2

up vote 3 down vote accepted

I recommend you take a look at Wilf's generatingfunctionology. It is the most intuitive and pedagogical place to learn about generating functions that I know about.

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The ordinary generating function for the sequence $a_0,a_1,a_2,\dots$ is $a_0+a_1x+a_2x^2+\cdots$. So the OGF for $1,0,1,0,\dots$ is $1+x^2+x^4+\cdots$.

The point is that you have replaced a sequence with a function, and you know lots of stuff about functions, like how to differentiate them and antidifferentiate them and multiply them by other functions and so on. So, you can use what you know about functions to get information about sequences.

If you are studying this in a class, then surely you have a text or some lecture notes wherein there will be many worked examples. People don't give exams on topics they have never discussed in class.

Perhaps you should adjust your approach to mathematics. Trying to learn a whole topic the day before the exam is suicide.

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