Need zeros of Hermite type 'e' polynomials of orders 6 and 8

• To: mathgroup at smc.vnet.net
• Subject: [mg5159] Need zeros of Hermite type 'e' polynomials of orders 6 and 8
• From: Michael Hucka <hucka at eecs.umich.edu>
• Date: Wed, 6 Nov 1996 01:34:02 -0500
• Organization: University of Michigan EECS, Ann Arbor, Mich., USA
• Sender: owner-wri-mathgroup at wolfram.com

```I have a problem in my engineering research for which I need to find the
zeros of the 6th and 8th Hermite polynomials.  The particular polynomials I'm
working with are the type 'e' polynomials He(x):

n       2                  2
He(x) = (-1)  Exp[ x/2 ] d/dx [ Exp[ -x/2 ] ]

although I'd settle for an answer for the more common Hermite Hn(x)
polynomials.

I can find the zeros for orders 1-5, but so far 6-8 have eluded me.  I've
attempted to get Mathematica to solve for this, but the results even for the
6th order are extremely long and are complex numbers, which I *think*
shouldn't be the case for the zeros of the Hermite polynomials.

I've tried looking in various references such as Abramowitz & Stegun, but it
appears there is no known expression for this.  Approximation formulas exist,
but since I need only the fairly low orders, and since the 5th order solution
is easily found, and there is a certain regularity to the polynomials, I was
hoping that there might be a way to get exact solutions to the 6th and 8th
order polynomials.

Can anyone offer any leads on this, or tricks to try in Mathematica?

--
Mike Hucka        hucka at umich.edu        http://ai.eecs.umich.edu/people/hucka
Ph.D. candidate, computational models of human visual processing, U-M AI Lab
UNIX admin & programmer/analyst, EECS Dept., University of Michigan

```

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