Landau-Kernel coefficients

Question

I want to determine the coefficients of a function but I'm not sure how. I have: $$\varphi_{n}(t) = c_{n}(1-t^2)^n$$ for $|t| \leq 1 $ and $\varphi_{n}(t) = 0$ elsewhere. I must find $c_{n}$, so that $$1 = \int_{\mathbb{R}}\varphi_{n}(t)dt $$ Then, \begin{align} 1 &= \int_{-1}^{1}c_{n}(1-t^2)^ndt \\ \frac{1}{c_{n}} &= \int_{-1}^{1}(1-t^2)^ndt \end{align} Is this right? And what do I do next, integration by parts? But when I integrate by parts it just becomes more complex. Or use the Binomial Theorem? I would appreciative some help :)

Answer 1

The best thing to do is to use integration by parts $n$ times.

$$\int_{-1}^1(1-x^2)^ndx=\int_{-1}^12n(1-x^2)^{n-1}x^2dx=\cdots=\int_{-1}^1\frac{2^kn(n-1)\cdots(n-k+1)}{\Pi_{j=1}^{n}(2j-1)}(1-x^2)^{n-k}x^{2k}dx.$$

Setting $k=n$ and integrating we get,

$$\frac{1}{c_n}=\frac{2^{n+1}n!}{\Pi_{j=0}^{n}(2j+1)}=\frac{2^{2n+1}(n!)^2}{(2n+1)!}.$$