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Keeping the Legendre polynomials in expressions without finding the explicit polinomials

• To: mathgroup at smc.vnet.net
• Subject: [mg57798] Keeping the Legendre polynomials in expressions without finding the explicit polinomials
• From: "Vladislav" <kazimir04 at yahoo.co.uk>
• Date: Thu, 9 Jun 2005 05:17:32 -0400 (EDT)
• References: <d7dp2r$qam$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi, all,

Thank you for your replies. I have learnt a lot of usefull things, but
Infortunately I had simplified my task very much, so that I could not
apply any answer directly to my case.

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. At the
same time 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,

Vlad