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MathGroup Archive 2006

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Re: Known recursion link to Hermite polynomials not solved in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69536] Re: Known recursion link to Hermite polynomials not solved in Mathematica
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Fri, 15 Sep 2006 06:44:46 -0400 (EDT)
  • References: <ee647q$5nb$1@smc.vnet.net> <eebd4g$l8m$1@smc.vnet.net>

Paul Abbott wrote:

>
>
> I don't think so. If you try
>
> ExponentialGeneratingFunction[{a[0] == 1, a[1] == 2,
> a[n] == a[n - 1] - (n - 1) a[n - 2]}, a[n], n, z]
>
> you will find that the solution involves Exp[-(z - 1)^2 / 2] -- which
> leads to HermiteH -- but that this multiplies
>
> Erfi[(z - 1)/Sqrt[2]]
>
> Cheers,
> Paul
>
> _______________________________________________________________________
> Paul Abbott Phone: 61 8 6488 2734
> School of Physics, M013 Fax: +61 8 6488 1014
> The University of Western Australia (CRICOS Provider No 00126G)
> AUSTRALIA http://physics.uwa.edu.au/~paul
>
Paul Abbott,
I've done some investigation of your solution.
With Bessel function recurrences  you are allowed negative and rational  
quantum
numbers, but not with HermiteH. What appears to happen is that the
normal integral function which is called the error function
behaves in a way to make this up for these negative or integral states.
The two types  ( Bessel and Hermite-error function) seem to be  mirrors 
of each other except for this.
1) The Hermite-error function type
a[n]=a0* a[n - 1] + (b0*n +c0) a[n - 2]: a0,b0,c0 Integers
2) The Bessel type is:
a[n]=(a0*n+b0) a[n - 1] + c0 a[n - 2]: a0,b0,c0 Integers

The result is more a reflection of the limitations of the current 
mathematical notation in Mathematica than
the actual sequential recursions.
It seems strange that the mathematics of probability  ( which is widely 
used in many fields)
should be less solvable ( Using the RSolve level of discrete functions)  
in Mathematica than that of
the Bessel function which isn't all that used in the real world.

As far as I know no one ( or can search)  has noticed or reported this 
type of recurrence
or this symmetry property before.
Roger Bagula


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