       Associated Legendre Function Problem in mma?

• To: mathgroup at smc.vnet.net
• Subject: [mg4361] Associated Legendre Function Problem in mma?
• From: siegman at ee.stanford.edu (A. E. Siegman)
• Date: Mon, 15 Jul 1996 05:01:00 -0400
• Organization: Stanford University
• Sender: owner-wri-mathgroup at wolfram.com

```Associated Legendre functions are a real bear.  Trying to
cope with them, I note in Abramowitz and Stegun, p. 334,
Eq. (8.6.16), that one of these functions which I
particularly need to use, LegendreP[n,-n,x], has the special
case (in TeX notation):

P_n^{(-n}(x) = 2^{-n} (x^2-1)^{n/2} / \Gamma[n+1]

But when I try to confirm this with mma, I see that the
magnitudes are OK, but there is a residual confusion

specialCase[n_,x_] := 2^(-n) (x^2-1)^(n/2) /
Gamma[n+1]

Table[{n, LegendreP[n,-n,x] /
specialCase[n,x] // Simplify},
{n,0,5}]  // TableForm

0   1

2
Sqrt[1 - x ]
-------------
2
1   Sqrt[-1 + x ]

2   -1

2
Sqrt[1 - x ]
-(-------------)
2
3     Sqrt[-1 + x ]

4   1

2
Sqrt[1 - x ]
-------------
2
5   Sqrt[-1 + x ]

and unfortunately getting the phase angles right is
important in my problem.  Who's correct here?

I want to evaluate very high order polynomicals (n > 50)
using rational fraction values of x for accuracy (which
seem to work pretty well).  But while LegendreP[2n, x],
which I also need to use, seems to run fine in this way,
LegendreP[n,-n,x] slows to a crawl for n > 20 or thereabouts
-- even though the polynomial expressions for the regular
and associated Legendre's are of the same order in the
two cases.  Hence the search for an alternative for the
associated case.

--AES   siegman at ee.stanford.edu

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```

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