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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
else please help me resolve this?
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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