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

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?

• Is this question related to Mathematica (TM)? Can you post your Mathematica code? Jul 2, 2013 at 17:24
• Have you seen this? functions.wolfram.com/HypergeometricFunctions/HermiteHGeneral/… Jul 2, 2013 at 17:30
• Related (duplicate?): mathematica.stackexchange.com/q/22040/5
– rm -rf
Jul 2, 2013 at 17:32
• @ bill s: as I understand, its Hermit polynomial-not function...
– Mack
Jul 2, 2013 at 17:40
• To the closers: I don't see any reason to close this as off topic. Mathematica has a function to deal with problems like this(Orthogonalize), so to me it looks like a fine Mathematica topic. Jul 2, 2013 at 20:22

## 1 Answer

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}
];