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Infinite polynomials

  1. Feb 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Is it possible to describe some infinite degree polynomials by their roots in a way analagous to finite degree polynomials?


    2. Relevant equations



    3. The attempt at a solution

    I know that, since not all infinite degree polynomials have roots (e.g. the power series representation of e^x), it would not be possible to do so for all of them. But what about polynomials like the power series of sin(x)? I was thinking maybe

    [tex]\prod^\infty_{n=0} \left ( x^2 - n^2\pi^2 \right )[/tex]
     
  2. jcsd
  3. Feb 5, 2008 #2

    Hurkyl

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    Staff Emeritus
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    Gold Member

    There's no such thing as an infinite degree polynomial. I presume you mean a power series.

    If (the analtyic continuation) of your power series is actually meromorphic, then there is a general factorization theorem. See:

    http://en.wikipedia.org/wiki/Infinite_product
     
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