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Functional equations for HermiteH[n,x]

• To: mathgroup at smc.vnet.net
• Subject: [mg58815] Functional equations for HermiteH[n,x]
• From: janostothmeister at gmail.com
• Date: Tue, 19 Jul 2005 04:10:33 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Hi, All,

1. I have found in the help that
â??_z HermiteH[n, z]
2 n HermiteH[-1+n,z]

Nice. I wanted to reproduce this myself.

FullForm[Hold[â??_z HermiteH[n, z]]]
Out[31]//FullForm=
Hold[D[HermiteH[n,z],z]]

Then, it should also work for me:
D[Hermite[n,z],z]

\!\(\*
  RowBox[{
    SuperscriptBox["Hermite",
      TagBox[\((0, 1)\),
        Derivative],
      MultilineFunction->None], "[", \(n, z\), "]"}]\)

But it does not.

2. I would also like to have H[n,-x]==-H[n,x],
but even FunctionExpand does not produce this.

3. This should be zero.
FunctionExpand[HermiteH[n + 1,
   x] - 2x HermiteH[n, x] + 2n HermiteH[n -
    1, x], n â?? Integers â?§ n > 0 â?§ x â?? Reals]

4. This is known to be zero:
Integrate[HermiteH[n, x] E^(-x^2, {x,-â??,â??},
Assumptions ->(n â?? Integers â?§ n > 0)]

5. This should be the KroneckerDelta[m,n]:
Integrate[HermiteH[n, x]HermiteH[m, x]E^(-x^2), {x, -â??, â??},
      Assumptions -> (n â?? Integers â?§ m â?? Integers â?§ n > 0 â?§ m
> 0)]

I know, I know, mathematical program packages know everything except
symbolic calculations, still...

Can anybody help me?

Thanks,

János

• Follow-Ups:
  • Re: Functional equations for HermiteH[n,x]
    • From: Pratik Desai <pdesai1@umbc.edu>