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