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Preventing LegendreP from self-extracting in manipulations

• To: mathgroup at smc.vnet.net
• Subject: [mg57829] Preventing LegendreP from self-extracting in manipulations
• From: "Vladislav" <kazimir04 at yahoo.co.uk>
• Date: Thu, 9 Jun 2005 05:55:30 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Hi, all,

I have already tried to post the message, but it went lost. I repeat it
as a new therad.

Can somebody help me with the following. I need present some functions
(prolate spheroidal functions) in the basis of the Legendre
polynomials. I.e.
I have functions like

FF1 = 0.6 LegendreP[5, #1] + 0.7 LegendreP[6, #1] &
FF2 = 0.3 LegendreP[5, #1] + 0.2 LegendreP[6, #1] &

I want to manipulate these functions and remain in the basis of prolate
functions.
For example I want to create a linear combination of functions, or
something like this.

FF = .2FF1[#1] + .3FF2[#1] &

It works well from the point of view of finding the numerical result,
but it do not give
the presentaion of the function in the basis of the Legendre
polynomials

I would like to have create a function which would give the result like

FFX = 0.9 LegendreP[5, #1] + 0.9 LegendreP[6, #1] &, so that I could
see the presentaion
of the function by typing FFX and obtaining  0.9 LegendreP[5, #1] + 0.9
LegendreP[6, #1] &.
In practice these functions contain much more terms and having the form
like
 0.9 LegendreP[5, #1] + 0.9 LegendreP[6, #1] & is very important. In
the same way
I would not like to have the explicit presentation as plolinomials,
like -0.28125+
1.6875  w + 5.90625 w^2 + .. because of loss of accuracy for future
results.

Sincerely,

Vladislav