91Ó°ÊÓ

Prove the double factorial identities: $$ \begin{gathered} (2 n) ! !=2^{n} n ! \\ (2 n-1) ! !=\frac{(2 n) !}{2^{n} n !} \end{gathered} $$

The Hermite polynomials, \(H_{n}(x)\), satisfy the following: i. \(=\int_{-\infty}^{\infty} e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\pi} 2^{n} n ! \delta_{n, m}\) ii. \(H_{n}^{\prime}(x)=2 n H_{n-1}(x)\). iii. \(H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x)\) iv. \(H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}}\left(e^{-x^{2}}\right) .\) ng these, show that a. \(H_{n}^{\prime \prime}-2 x H_{n}^{\prime}+2 n H_{n}=0\). [Use properties ii. and iii.] b. \(\int_{-\infty}^{\infty} x e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\pi} 2^{n-1} n !\left[\delta_{m, n-1}+2(n+1) \delta_{m, n+1}\right] .\) [Use properties i. and iii.]

Show that a Sturm-Liouville operator with periodic boundary condion \([a, b]\) is self-adjoint if and only if \(p(a)=p(b)\). [Recall that periodic dary conditions are given as \(u(a)=u(b)\) and \(\left.u^{\prime}(a)=u^{\prime}(b) .\right]\).