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Thread: Help with exponential generating function proof

  1. #1
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    Help with exponential generating function proof

    Use exponential generating functions to prove that:
    P(n) = P(n-1) + n, with n >= 1 and P(0) = 1
    will result in this equation: P(n) = (1/2)n2 + (1/2)n + 1

    I know how to do this with generating functions, you just plug in n=1, n=2, n=3,....
    P(1) will result in (1/2)(1)2 + (1/2)(1) +1 which is 2
    P(2) will result in P(2-1)+2 which is P(1) + 2.. and so on
    but I'm kind of stuck on "exponential" generating functions.
    Can someone help me out?
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  2. #2
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    Re: Help with exponential generating function proof

    Quote Originally Posted by chinknig View Post
    Use exponential generating functions to prove that:
    P(n) = P(n-1) + n, with n >= 1 and P(0) = 1
    will result in this equation: P(n) = (1/2)n2 + (1/2)n + 1

    I know how to do this with generating functions, you just plug in n=1, n=2, n=3,....
    P(1) will result in (1/2)(1)2 + (1/2)(1) +1 which is 2
    P(2) will result in P(2-1)+2 which is P(1) + 2.. and so on
    but I'm kind of stuck on "exponential" generating functions.
    Can someone help me out?
    First here is a good FREE source textbook.

    Now, just from reading your post, I for one do not understand what is required.
    It seems as if you placed us into the middle of some working.
    May I suggest that you give the context or setting along with the complete and exact wording of the exercise.
    Thanks from topsquark
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  3. #3
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    Re: Help with exponential generating function proof

    the exponential generating function is

    f(x)=\sum _{n=0}^{\infty } \frac{p_n}{n!}x^n

    compute

    f'(x)-f(x)=e^x+x e^x

    more computation

    f(x)=\frac{1}{2}e^x\left(x^2+2x+2\right)

    more computation
    Thanks from topsquark
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  4. #4
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    Re: Help with exponential generating function proof

    I've proven how to derive the second equation from the eqution before using generating functions. That's why I know that the answer should be P(n) = (1/2)n^2 + (1/2)n + 1.
    But know the instructor wants the class to use 'exponential' generating functions.
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  5. #5
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    Re: Help with exponential generating function proof

    Quote Originally Posted by Idea View Post
    the exponential generating function is

    f(x)=\sum _{n=0}^{\infty } \frac{p_n}{n!}x^n

    compute

    f'(x)-f(x)=e^x+x e^x

    more computation

    f(x)=\frac{1}{2}e^x\left(x^2+2x+2\right)

    more computation
    Could you show more steps? I'm trying to follow what you're showing me.
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  6. #6
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    Re: Help with exponential generating function proof

    First, solve the differential equation to get f(x)

    Next,

    f(x)=\left(\frac{1}{2}x^2+x+1\right)e^x

    =\left(1+x+\frac{1}{2}x^2\right)\left(1+\frac{x}{1  !}+\frac{x^2}{2!}+\text{...}\right)

    =1 + 2x +\text{...}+ \left(\frac{1}{n!}+\frac{1}{(n-1)!}+\frac{1/2}{(n-2)!}\right)x^n+\text{...}

    so that

    \frac{p_n}{n!}=\frac{1}{n!}+\frac{1}{(n-1)!}+\frac{1/2}{(n-2)!}

    and

    p_n=1+\frac{n}{2}+\frac{n^2}{2}
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