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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,
JÃ¡nos
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