       Re: Functional equations for HermiteH[n,x]

• To: mathgroup at smc.vnet.net
• Subject: [mg58828] Re: [mg58815] Functional equations for HermiteH[n,x]
• From: Pratik Desai <pdesai1 at umbc.edu>
• Date: Wed, 20 Jul 2005 00:29:25 -0400 (EDT)
• References: <200507190810.EAA11598@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```janostothmeister at gmail.com wrote:

>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//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.
>
>
If you try this it seems to work
D[HermiteH[n, z], {z, 1}]
>>2 n HermiteH[-1 + n, z]

>2. I would also like to have H[n,-x]==-H[n,x],
>but even FunctionExpand does not produce this.
>
>
But this does work, I dont know if they mean the same thing
HermiteH[2, z] == HermiteH[2, -z]
>>True

>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]
>
>
n = 2
HermiteH[n + 1, x] - 2x HermiteH[n, x] + 2n HermiteH[n - 1, x] // Simplify
>>0
I get the feeling that Functionexpand does not work for this function

>4. This is known to be zero:
>Integrate[HermiteH[n, x] E^(-x^2, {x,-â??,â??},
>Assumptions ->(n â?? Integers â?§ n > 0)]
>
>
Integrate[HermiteH[n, x] E^(-x^2), {x, -Inf, Inf}]
>>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)]
>>
>>
>
>
>
How about if you try this
Table[Integrate[HermiteH[n, x]*
HermiteH[m, x]*E^(-x^2), {x, -Inf, Inf}], {m, 1, 4, 1}, {n, 1, 4,
1}] // MatrixForm

>I know, I know, mathematical program packages know everything except
>symbolic calculations, still...
>
>Can anybody help me?
>
>Thanks,
>
>JÃ¡nos
>
>
>
Best regards

Pratik

--
Pratik Desai