Gram-Schmidt process with Hermite functions on [-1, 1]

Question

Denote by $h_n$ the $n$-th Hermite function. $$ h_n(x) = \frac{(-1)^n }{\sqrt{2^n n! \sqrt{\pi}}} \mathrm{e}^{\frac{x^2}{2}} \frac{\mathrm{d}^n}{\mathrm{d} x^n} \mathrm{e}^{-x^2} $$

I am trying to find the 40th, 41st and 42nd terms in Gram-Schmidt process with Hermite functions $h_n$ on $[-1, 1]$.

I've used the usual procedure for Gram-Schmidt process, but I've been able to calulate only the first 6 terms and then my computer got stuck.

Is there some way to calculate them?

Answer 1

Based on the Wikipedia article you referred, here is a code

u[1] = HermiteH[1, x];
Do[
  u[n] = HermiteH[n, x] - Sum[Integrate[HermiteH[n, x] u[i], 
  {x, -1.0, 1.0}], {i, 1,n-1}], {n, 2, 45}
];