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.]