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MathGroup Archive 1997

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Fwd: Pls help me translate Laguerre poly to Hermite type "Hn" in MMA 2.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg5705] Fwd: [mg5683] Pls help me translate Laguerre poly to Hermite type "Hn" in MMA 2.
  • From: BobHanlon at aol.com
  • Date: Sat, 11 Jan 1997 14:29:04 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

To be a polynomial, the first parameter in LaguerreL must be an integer.  For

non-integer values of the first parameter, LaguerreL is the generalized 
Laguerre function.

The transformation which you used is only valid for restricted values of the 
parameters and these restrictions should be included in the transformation's 
definition:

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

Comparing the first several values:

Table[LaguerreL[n, -1/2, x] == lag[n, -1/2, x], 
	{n, 0, 7}] // Simplify

{True, True, True, True, True, True, True, True}

The generalized Laguerre function is equivalent to the confluent 
hypergeometric function.  The transformation is (generalization of A&S 
22.5.54)

lag[nu_, a_, x_] := Binomial[nu + a, nu] * 
	Hypergeometric1F1[-nu, a + 1, x]

lag[1/2, -1/2, 1]

                     1
Sqrt[Pi] Binomial[0, -] 
                     2
 
                                 1   1
  Hypergeometric1F1Regularized[-(-), -, 1]
                                 2   2

% /. Binomial[x_, y_] -> Gamma[x+1]/(Gamma[y+1] * 
	Gamma[x-y+1])

                                 1   1
2 Hypergeometric1F1Regularized[-(-), -, 1]
                                 2   2
------------------------------------------
                 Sqrt[Pi]

% == LaguerreL[1/2, -1/2, 1]

True

A&S 13.6.15 would enable you to express this specific confluent hypergeometic

function in terms of the Weber (parabolic cylinder) function.

In version 3.0 both lag[1/2, -1/2, 1] and LaguerreL[1/2, -1/2, 1] result in 

2 (E - Sqrt[Pi] Erfi[1])/Pi

---------------------
Forwarded message:
From:	hucka at eecs.umich.edu (Michael Hucka)
To: mathgroup at smc.vnet.net
To:	mathgroup at smc.vnet.net

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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