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