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Math Help - Sequence question

  1. #1
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    Sequence question

    Hi, I am currently stuck on this sequence question, I tried playing around with it for a while now, can't seem to solve it. Any help would be appreciated!

    What are a necessary and sufficient set of conditions on these integer sequences \{a_n\}_{n \ge 0} and \{b_n\}_{n \ge 0} in order to satisfy the equation A(x) \times B(x) = 1 for all x \in \mathbb{R} such that A(x), B(x) are defined. Where A(x) = a_0+a_1x+a_2x^2+ \cdots and B(x) = b_0+b_1x+b_2x^2+ \cdots

    Thanks again!
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  2. #2
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  3. #3
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    Thanks Opalg, excellent explanations!

    Also I'm stuck on another extension of this problem. It says "Assuming A(x) \times B(x) = 1, find a recurrence formula for the integer sequence {b_n} in terms of the integer sequence {a_n}

    Now I've gotten this far:

    a_0b_0 = 1 so b_0 = \frac{1}{a_0}

    By the same principles b_2 = \frac{a_1^2a_0-a_2}{a_0^2}

    But as you keep going b_3, b_4 get extremely messy and I can not find any relationship or pattern between the b_n's and a_n's.

    Could anyone shed some light?

    Thanks!
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  4. #4
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    Quote Originally Posted by usagi_killer View Post
    Thanks Opalg, excellent explanations!

    Also I'm stuck on another extension of this problem. It says "Assuming A(x) \times B(x) = 1, find a recurrence formula for the integer sequence {b_n} in terms of the integer sequence {a_n}
    I read this and gave it a go but couldn't get far. I see what you mean by it getting very messy. I am sorry I couldn't help but now I really want to find out how this is done.
    Last edited by Nguyen; September 1st 2010 at 04:15 AM.
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  5. #5
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    I think that the only thing resembling a recurrence relation that is satisfied by these coefficients is the relation \sum_{k=0}^na_kb_{n-k} = 0, which implies that b_n = -a_0^{-1}\sum_{k=1}^na_kb_{n-k}. That hardly counts as recurrence relation in the usual sense, because the number of terms in the relation varies with n.

    Even for a function like \cosh x, which has a very simple power series, the coefficients of the reciprocal function 1/\cosh x are not easy to determine. In fact, they involve the Euler numbers, which do not satisfy any straightforward recurrence relation.
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  6. #6
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    An additional question which follows from the previous questions.
    If for all n≥0 and , then what are the ?

    Thanks for the help.
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