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November 8th, 2013, 06:24 PM   #1
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Coefficient of Generating Function

Hello, practicing for my upcoming exam and there's a question that I'm not familiar with the method to do it.

Given the generating function : P(x) = (2-5x)/(1-5x+6x^2), how do we find an expression of Pn in terms of past values of Pn?

Help please!!!
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November 8th, 2013, 06:43 PM   #2
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Re: Coefficient of Generating Function

I don't know what method you might have been taught, but let me see what I can do.

The generating function is rational and its denominator is of degree 2, so there is a 2-term recurrence relation. That is, you can write P_{n} = a*P_{n-1} + b*P_{n-2} for some constants a and b. So just find the first few terms of the sequence and solve the resulting system of equations.
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November 8th, 2013, 06:51 PM   #3
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Re: Coefficient of Generating Function

I'm having trouble with finding the first coefficients.
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November 8th, 2013, 07:12 PM   #4
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Re: Coefficient of Generating Function

The terms of the generating function?
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November 8th, 2013, 07:59 PM   #5
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Re: Coefficient of Generating Function

yes
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November 9th, 2013, 08:53 AM   #6
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Re: Coefficient of Generating Function

It's just long division.

Code:
            2 + 5x + ...
          _________
1-5x+6x^2 ) 2-5x
           -2+10x-12x^2
           ------------
               5x-12x^2
              -5x+25x^2-30x^3
               --------------
               ...
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