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

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

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)

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
 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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