Proving Hermite Equation with Hermite Polynomials

[No Name Required]

Im stuck on this question :(


The Hermite polynomials can be defined through

$$\displaystyle{F(x,h) = \sum^{\infty}_{n = 0} \frac{h^n}{n!}H_n(x)}$$

Prove that the $$H_n$$ satisfy the hermite equation

$$\displaystyle{H''_n(x) - 2xH'_n(x) + 2nH_n(x) = 0}$$

Using

$$\displaystyle{\sum^{\infty}_{n = 0} \frac{h^n}{n!}nH_n(x) = h\frac{\partial}{\partial h}F(x,h)}$$

Can someone give me a bit of a push in the right direction?

 

[dextercioby]

Who's $F(x,h)$ ...?

IIRC, it has to have some exponential, right...?

Daniel.

 

[HallsofIvy]

Im stuck on this question :(


The Hermite polynomials can be defined through

$$\displaystyle{F(x,h) = \sum^{\infty}_{n = 0} \frac{h^n}{n!}H_n(x)}$$

Prove that the $$H_n$$ satisfy the hermite equation

$$\displaystyle{H''_n(x) - 2xH'_n(x) + 2nH_n(x) = 0}$$

Using

$$\displaystyle{\sum^{\infty}_{n = 0} \frac{h^n}{n!}nH_n(x) = h\frac{\partial}{\partial h}F(x,h)}$$

Can someone give me a bit of a push in the right direction?

What's wrong with "just do it!"? Just go ahead and differentiate that sum, term by term, (your hint essentially says that you can do that), plug into the differential equation and see what happens.