By definition, the Hermite Polynomials, H_n(x), are the coefficients of the Taylor expansion of the above function around zero. Thus it is easy to obtain them:
So the first four Hermite polynomials are:
Now we can compute:
So the functions Y_n(x) are in L^2(R) at least for the first few values of n. In class we will prove that Y_n(x) are in L^2(R) for every n.
Recall that the Legendre functions may be obtained by orthonormalizing the functions x^n, see the first handout. Hermite polynomials may be obtained in a similar way.
Next, load in the appropriate package
Now we can use the Gram-Schmidt process to orthonrmalize the functions a_n(x)
So you can decide what the relationship between these functions and H_n(x) is.
Calculating the inner products <Y_n(x),Y_m(x)> are easy. First define
So as you can see these eigenvalues are orthogonal.