Proving A_n(r) with Integer Coeff Polynomials: Q&A

[physicsRookie]

Define a sequence
$$A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.$$

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.

 

[matt grime]

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