Chapter 16, Problem 19PS

(a)

To determine

To calculate: The solution of the Hermite’s equation, $y\text{'}\text{'}-2xy\text{'}+2ky=0$ when $k=4$, using a power series.

(b)

To determine

The values of Hermite polynomials $H_0\left(x\right),H_1\left(x\right),H_2\left(x\right),H_3\left(x\right)\text{, and }{H}_4\left(x\right)$ if $H_k\left(x\right)$ is given by $H_k\left(x\right)={\displaystyle \sum _{n=0}^P\frac{{\left(-1\right)}^{n}k!{\left(2x\right)}^{k-2n}}{n!\left(k-2n\right)!}}$.