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?

  • $\begingroup$ Is this question related to Mathematica (TM)? Can you post your Mathematica code? $\endgroup$ Jul 2, 2013 at 17:24
  • $\begingroup$ Have you seen this? functions.wolfram.com/HypergeometricFunctions/HermiteHGeneral/… $\endgroup$
    – bill s
    Jul 2, 2013 at 17:30
  • $\begingroup$ Related (duplicate?): mathematica.stackexchange.com/q/22040/5 $\endgroup$
    – rm -rf
    Jul 2, 2013 at 17:32
  • $\begingroup$ @ bill s: as I understand, its Hermit polynomial-not function... $\endgroup$
    – Mack
    Jul 2, 2013 at 17:40
  • 4
    $\begingroup$ 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. $\endgroup$ Jul 2, 2013 at 20:22

1 Answer 1


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

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

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