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 Post subject: Limit of a Generating FunctionPosted: Sun, 31 Jan 2010 05:32:42 UTC
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Joined: Sun, 31 Jan 2010 05:07:26 UTC
Posts: 88
Some background in generating functions might be helpful.

Is it possible to use a limit operation or equivalent to extract coefficients for recursive use inside of a generating function?

For example, suppose there is an ogf:
gf = c1 + c2x + ... cnx^n + ... + c100x^100

We could use such an operation to eventually extract the middle in terms of a defined function

a = limit[gf, x->0]
b = limit[gf/(x^100), x->inf.]

c = gf + (gf*y- a*y - b*yx^100)

This should start a second series that contains successively smaller portions of the original gf until only the middle element remains. However, the extraction of the remainder is not defined yet. Is it possible to use mathematics, rather than a computer, to extract the remaining values one at-a-time?

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 Post subject: Posted: Mon, 8 Feb 2010 18:41:13 UTC
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Joined: Tue, 20 Nov 2007 04:36:12 UTC
Posts: 854
Location: Las Cruces
To make this problem understandable, I think that you must specify which of the symbols are "given" and which are "unknown". Are you assuming that the are unknown but that the results of the limit operations such as are known?

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 Post subject: Clarification...Posted: Thu, 18 Feb 2010 07:11:33 UTC
 Senior Member

Joined: Sun, 31 Jan 2010 05:07:26 UTC
Posts: 88
Only the function for the entire generating function is known. However, the coefficients for the g.f. start with x^0 and proceed uniformly to x^(2k) for some k. How is the formula for the central element recovered?

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 Post subject: Re: Clarification...Posted: Thu, 18 Feb 2010 15:07:51 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7624
Location: NCTS/TPE, Taiwan
Generator wrote:
Only the function for the entire generating function is known. However, the coefficients for the g.f. start with x^0 and proceed uniformly to x^(2k) for some k. How is the formula for the central element recovered?

Hint:
Spoiler:
What do you get when you differentiate gf n-times and set x=0?

(Edit: use hide tags...)

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Last edited by outermeasure on Thu, 18 Feb 2010 16:21:49 UTC, edited 1 time in total.

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 Post subject: Posted: Thu, 18 Feb 2010 16:18:25 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15563
Location: Austin, TX
For proposed problems, please use the hide tags for hints.

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