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MathGroup Archive 2005

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Re: Functional equations for HermiteH[n,x]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58827] Re: Functional equations for HermiteH[n,x]
  • From: dh <dh at metrohm.ch>
  • Date: Wed, 20 Jul 2005 00:29:24 -0400 (EDT)
  • References: <dbie83$bt0$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Janos,
Mathematica does not know too man features about Hermite polynomials. 
But needed features can be added. See below

sincerely, Daniel

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[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.
Yes it does if you write HermiteH instead of Hermite
> 
> 2. I would also like to have H[n,-x]==-H[n,x],
> but even FunctionExpand does not produce this.
This can be done by adding a rule to HermiteH.
First you must unprotect HermiteH:
Unprotect[HermiteH]
then you can define:
HermiteH[n_, -x] := -HermiteH[n, x]

> 
> 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]
This can be achieved by defining a rule for H[n+1,x] as above:
HermiteH[n_Integer+1,x_]:= 2 x HermiteH[n,x]-2 n HermiteH[n-1,x]
> 
> 4. This is known to be zero:
> Integrate[HermiteH[n, x] E^(-x^2, {x,-â??,â??},
> Assumptions ->(n â?? Integers â?§ n > 0)]
A further rule is needed here.
You would assigne this rule not to Integrate, but to HermiteH by:
HermiteH/; .....
> 
> 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)]
Same thing as 4)
> 
> 
> I know, I know, mathematical program packages know everything except
> symbolic calculations, still...
> 
> Can anybody help me?
> 
> Thanks,
> 
> János
> 


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