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

• To: mathgroup at smc.vnet.net
• Subject: [mg69583] Re: Known recursion link to Hermite polynomials not solved in Mathematica
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 16 Sep 2006 03:50:22 -0400 (EDT)
• Organization: The University of Western Australia
• References: <ee647q$5nb$1@smc.vnet.net> <eebd4g$l8m$1@smc.vnet.net> <eee0d6$6f3$1@smc.vnet.net>

In article <eee0d6$6f3$1 at smc.vnet.net>,
 Roger Bagula <rlbagula at sbcglobal.net> wrote:

> 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.

This is not clear to me -- nor I expect to most readers of this group.

> 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

To be consistent, surely you should have

 a[n]=(b0*n+a0) a[n - 1] + c0 a[n - 2]: a0,b0,c0 Integers

so that the relationship between the recurrences is clear.

> The result is more a reflection of the limitations of the current 
> mathematical notation in Mathematica than the actual sequential 
> recursions.

What limitation of mathematical notation in Mathematica?

> 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.

What on earth do you mean? The Bessel function appears everywhere in the 
real world! One nice example: the natural shape of a horn is a bessel 
function. See

 http://www-ccrma.stanford.edu/~jos/tiirts/Horn_Reflectance_Filter.html

And what about waves in 2D, and normal modes of a drum?

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

If you can more clearly describe the symmetry property you see, I expect 
that you will find it is well known ...

Cheers,
Paul

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Paul Abbott                                      Phone:  61 8 6488 2734
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