Extracting Information From Generating Functions

Generator · **Joined:** Sun, 31 Jan 2010 05:07:26 UTC **Posts:** 88

I'm attempting to work with generating functions that are relatively simple to work with in what I believe is "closed form"; that is, the equation that represents the function. However, the coefficients are relatively complex and difficult to work with.

My problem is that I'd like to find the limit as the index of the coefficient approaches infinity, or one coefficient that is very deep into the equation. In other words, I'd like to find like the millionth term, or I'd also be even happier with the term that represents infinity. However, like I said, I'm having trouble extracting the individual coefficients, and it may be almost impossible to do so.

So I wonder if there is a method or methods that can extract one coefficient. My guess is that I might be able to use the limit to do so.

Example
For the generating function $\displaystyle\frac{1}{1-\frac{x}{2}}$ the inifinite term approaches

.

Please help me find the infinite or near infinite terms!

Shadow · **Posted:** Mon, 5 Jul 2010 18:26:10 UTC

Generator wrote:

I'm attempting to work with generating functions that are relatively simple to work with in what I believe is "closed form"; that is, the equation that represents the function. However, the coefficients are relatively complex and difficult to work with.

My problem is that I'd like to find the limit as the index of the coefficient approaches infinity, or one coefficient that is very deep into the equation. In other words, I'd like to find like the millionth term, or I'd also be even happier with the term that represents infinity. However, like I said, I'm having trouble extracting the individual coefficients, and it may be almost impossible to do so.

So I wonder if there is a method or methods that can extract one coefficient. My guess is that I might be able to use the limit to do so.

Example
For the generating function $\displaystyle\frac{1}{1-\frac{x}{2}}$ the inifinite term approaches

.

Please help me find the infinite or near infinite terms!

Can you give the specific function you're working with whose coefficients are "complex and difficult to work with"?

What are you trying to find the limit of as your index approaches infinity?

You do realize that convergence of the function isn't really necessary for most of the things you care about, right? Furthermore if your function does converge then the nth term HAS to go to zero by the nth term test for convergence.

Lastly, if you're dealing with "the generating function for f(x)" then the millionth term is just f(1.000.000).

If you're still having trouble, please say so, and please be specific so that it'll be easier to help you.

Generator · **Joined:** Sun, 31 Jan 2010 05:07:26 UTC **Posts:** 88

Actually, I got a general idea of operating on series, and I think that working with limits will help solve my problem. I posted a similar question about SERIES as opposed to GENERATING FUNCTIONS here and realized that treating the function as a series will work better.

I'm actually wondering about the millionth coefficient, and not the function of one million. I guess I realized that what I'm working with is more of a series than a generating function, although operations on generating functions are coming in handy. I guess I was figuring that since generating functions can be used for statistics, there might be an easy way using that methodology to extract a given coefficient.

I wanted to thank you for your help, though. You guys have really given me some good ideas and helped me to get a better understanding of the math. :-)

Shadow · **Posted:** Wed, 7 Jul 2010 00:20:42 UTC

Generator wrote:

Actually, I got a general idea of operating on series, and I think that working with limits will help solve my problem. I posted a similar question about SERIES as opposed to GENERATING FUNCTIONS here and realized that treating the function as a series will work better.

I'm actually wondering about the millionth coefficient, and not the function of one million. I guess I realized that what I'm working with is more of a series than a generating function, although operations on generating functions are coming in handy. I guess I was figuring that since generating functions can be used for statistics, there might be an easy way using that methodology to extract a given coefficient.

I wanted to thank you for your help, though. You guys have really given me some good ideas and helped me to get a better understanding of the math. :-)

You're confusing the generating function with the function that the generating function generates.

If I put

$$g(x)=\sum_{n=0}^\infty f(n)x^n$

then I mean that g is the (ordinary) generating function for f.

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Extracting Information From Generating Functions

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