Sir Harold Jeffreys wrote: $\dagger$
Consider the integrals $$ I_n = \int_{-1}^1 (1-x^2)^n \cos\alpha x \,dx. $$ Two integrations by parts give the recurrence relation $$ \alpha^2 I_n = 2n(2n-1) I_{n-1} - 4n(n-1) I_{n-2}, \qquad n\ge 2. $$
So I did the two integrations by parts and got $$ \frac{2n}{\alpha^2} \int_{-1}^1 \cos(\alpha x) (1-x^2)^{n-2}(1 - (2n-1) x^2) \, dx, $$ so I had to ascertain whether that was equal to the right side of the recurrence relation. That comes down to this:
\begin{align} & (1-x^2)^{n-2}(1 - (2n-1) x^2) \\[6pt] = {} & (2n-1)\,\underbrace{(1-x^2)^{n-1}} {} - 2(n-1) \underbrace{(1-x^2)^{n-2}} \end{align} So my question is this: Is there some context in which it is standard to use the set $\{ (1-x^2)^n : n=0,1,2,\ldots \}$ as a basis of the space of even polynomials? What context would that be? In other words, does this fit neatly into some broader context of which this is a routine instance?
$\dagger$ Scientific Inference, third edition, Cambridge University Press, 1973. Appendix III. If I'm not mistaken, this appendix does not appear in other editions.