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 about phase angles: 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? Addendum: The reason for worrying about this is that 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 ==== [MESSAGE SEPARATOR] ====