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Polynomial recurrence equation

  1. Dec 29, 2004 #1
    For our combinatorics class there is a bonus excercise in which we have to solve the following recurrence relation:

    [tex]L_k(x) = L_{k-1}(x) + xL_{k-2}(x)[/tex]

    My first thought was to construct the following generating function:

    [tex]F(x,y) = \sum_{k=0}^{\infty} L_k(x) y^k[/tex].

    By putting in the recurrence relation I found a formula for this generating function:

    [tex]F(x,y) = \frac{1+xy}{1-y-xy^2}[/tex]

    Using the geometric series I found that this equals to:

    [tex]F(x,y) = (1+xy)\sum_{k=0}^{\infty} (y+xy^2)^k
    = \sum_{k=0}^{\infty} (1+xy)(xy^2)^k\frac{(1+xy)^k}{(xy)^k}
    = \sum_{k=0}^{\infty}\sum_{n=0}^k{k\choose n}(1+xy)x^ny^{n+k}[/tex]

    Now I've tried all afternoon to rewrite this whole powerseries in terms of y, so I can equate the generating function with this powerseries and find the required [tex]L_k(x)[/tex], but I could'nt do it :(

    Could someone give me a hint in which direction to look for now?

    Thanks in advance!!
     
  2. jcsd
  3. Dec 29, 2004 #2

    Hurkyl

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    Reverse the order of summation. (the idea is analogous to reversing the order of integration in an iterated integral)
     
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