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Define a sequence
A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.<br />
Prove that
A_n(r) = \frac{n!}{r^{2n+1}}[P_n(r)\sin r - Q_n(r)\cos r]
where P_n and Q_n are two polynomials with integer coefficients. What is the degree of P_n and of Q_n?
Can anyone help me? Thanks.
A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.<br />
Prove that
A_n(r) = \frac{n!}{r^{2n+1}}[P_n(r)\sin r - Q_n(r)\cos r]
where P_n and Q_n are two polynomials with integer coefficients. What is the degree of P_n and of Q_n?
Can anyone help me? Thanks.
