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How to prove that?

  1. Feb 25, 2006 #1
    Define a sequence
    [tex]A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.

    Prove that
    [tex]A_n(r) = \frac{n!}{r^{2n+1}}[P_n(r)\sin r - Q_n(r)\cos r] [/tex]

    where [tex]P_n[/tex] and [tex]Q_n[/tex] are two polynomials with integer coefficients. What is the degree of [tex]P_n[/tex] and of [tex]Q_n[/tex]?

    Can anyone help me? Thanks.
  2. jcsd
  3. Feb 25, 2006 #2

    matt grime

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    Science Advisor
    Homework Helper

    Induction. And an educated guess for the degrees, followed by induction.
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