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From: Axel Vogt <&noreply@axelvogt.de>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: Orthogonal Polynomials
Date: Sat, 09 Jul 2011 16:35:48 +0200
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On 08.07.2011 19:41, Mate wrote:
> Hello to all,
>
> I'd like to share here a recent experience in using
> Maple in a project requesting the manipulation of some
> orthogonal polynomials. It was a rather frustrating one
> and I mention it hoping to be useful for other users and also for
> Mapelsoft's stuff (if they read this forum) in order to do
> something about this.
>
> The OthogonalSeries package is not new in Maple, so

... (snipped to shorten for my reply)

> I have also found that some orthogonal functions are not well
> integrated into the system.
> For example, the obvious:
>
> int(ChebyshevT(40,x)^2*(1-x^2)^(-1/2),x=-1..1);
>
> cannot be computed to Pi/2 (at least in a reasonable time); it works
> however if the polynomial is expanded, or if its degree is e.g. 10.
>
> Finally the package was useful, but I would be interested if other
> users had similar experiences and if there is a better approach
> in such cases.
>
> Mate

I only used it occasionally and then only for computing
a base change.

For the above integral I could imagine, that Maple uses some
general representation for ChebyshevT (hypergeom 2F1 ?) and
can not find a solution, even for the specific situation (but
after 'expand' becomes aware of it).

For that I used Cheb:= (n,x) -> cos(n*arccos(x)) and then
   Int(ChebyshevT(40,x)^2*(1-x^2)^(-1/2),x=-1..1);
   subs(ChebyshevT = Cheb, %);
   value(%);
is quite fast.

However for orthogonality Cheb(m,x), Cheb(n,x) some care is
needed, M15 seems to ignore the special case m=n.

There is a package by Sergey Moiseev at Maple's Application
Center, which may be of interest for you.

Axel