Pls help me translate Laguerre poly to Hermite type "Hn" in MMA 2.2

• To: mathgroup at smc.vnet.net
• Subject: [mg5683] Pls help me translate Laguerre poly to Hermite type "Hn" in MMA 2.2
• From: Michael Hucka <hucka at eecs.umich.edu>
• Date: Tue, 7 Jan 1997 11:23:01 -0500
• Organization: University of Michigan EECS, Ann Arbor, Mich., USA
• Sender: owner-wri-mathgroup at wolfram.com

```I have a certain expression in my work that Mathematica 2.2 simplifies to
something containing a generalized Laguerre polynomial (LaguerreL).  The form
of the Laguerre is one with n = 1/2, a = -1/2, which MMA 2.2 apparently won't
express in simpler terms than, for example,

In[66]:= LaguerreL[1/2, -1/2, x]

1    1
Out[66]= LaguerreL[-, -(-), x]
2    2

I need to simplify my original expression further.  We don't have Mathematica
3.0 yet, only version 2.2, which doesn't have the FunctionExpand command,
which is what I really need for my problem.  So I'm trying to express the
generalized Laguerre in terms of another polynomial.  My copy of Abramowitz
and Stegun _Handbook of Mathematical Functions_ says that the generalized
Laguerre polynomial where a = -1/2 can be expressed as follows:

n
(-1/2)       (-1)         _
L     (x) = -------  H  (\/x )
n               2n   2n
n! 2

where H is the Hermite type "n" polynomial.  So I've tried the following
definition in Mathematica 2.2:

lag[n_, a_, x_] := (((-1)^n)/(n! * 2^(2*n))) * HermiteH[2*n, Sqrt[x]]

This seems right to me, but it does not seem to yield the same answers as
LaguerreL[n, -1/2, x].  For example:

In[74]:= LaguerreL[1/2, -1/2, 1]

1   1
2 Hypergeometric1F1Regularized[-(-), -, 1]
2   2
Out[74]= ------------------------------------------
Sqrt[Pi]

In[75]:= lag[1/2, -1/2, 1]

2 I
Out[75]= --------
Sqrt[Pi]

In[76]:= N[%%]

Out[76]= -0.131794

In[77]:= N[%%]

Out[77]= 1.12838 I

I must be doing something very wrong, but I just can't see it.  Can someone

In addition, if someone has access to MMA 3.0 and would run FunctionExpand on
LaguerreL[1/2, -1/2, x] and send me the result, I'd appreciate it.

--
Mike Hucka     hucka at umich.edu     <URL: 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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