Chapter 12.22, Problem 8P

In the generating function (22.16), expand the exponential in a power series and collect powers of h to obtain the first few Hermite polynomials. Verify the identity $\frac{{\partial }^2\Phi }{\partial x^2}-2x\frac{\partial \Phi }{\partial x}+2h\frac{\partial \Phi }{\partial h}=0$

Substitute the series in (22.16) into this identity to prove that the functions $H_n(x)$ in (22.16) satisfy equation (22.14). Verify that the highest term in $H_n(x)$ in (22.16) is ${(2x)}^n$.

[You have then proved that the functions called $H_n(x)$ are really theHermite polynomials since, by Problem 5, (22.14) has just one polynomial solution of degree n.]