Show that the Hermite polynomials H2(x) and H3(x).

[ASIWYFA]

Hi guys. I am new, and i need help badly. I have been asked this question and I have no idea how to do it. Any help would be appreciated!



Show that the Hermite polynomials H2(x) and H3(x) are orthogonal on
x € [-L, L], where L > 0 is a constant,
H2(x) = 4x² - 2 and H3(x) = 8x³ - 12x

Thanks in advance.

 

[micromass]

You need to show that

$$\int_{-L}^L{(4x^2-2)(8x^3-12x)dx}=0$$

This seems an easy integral...

Or do you consider another inner product??

 

[ASIWYFA]

Yes you are correct. THanks you. I believe I have solved the problem correctly.

 

[uart]

One thing that's worth pointing out about this problem is that while there are other ways for functions to be orthogonal over a specific interval (like [-1,1] for example), the only way possible for functions to be orthogonal over the arbitrary interval [-L,L] is if their product is an odd function.

So the original question is equivalent to showing that the product of H2 and H3 is an odd function.