From: Johannes H Andersen <johs@madasafish.com>
Subject: Re: Polynomial basis functions on the interval [0,1]
Date: Sat, 25 Mar 2000 23:09:50 +0000
Newsgroups: sci.math
Summary: [missing]

Les Wright wrote:
> 
> Hiram Berry wrote:
> 
> > Thankyou for the answer and reference, Johannes.  That's just the type of
> > answer I was looking for; in the meantime, while staring for a while at the
> > sequence of polynomials previously found, I did finally see the pattern for
> > recursively generating the particular sequence I posted previously:
> >
> > call the integer coefficient (excluding the sqrt part) of the m-order term
> > in the n-order polynomial Pn previously defined, "T(n,m)", then
> >
> > T(0,0) = 1
> > T(n,0) = -1 * T(n-1,0), n > 0
> > T(n,m) = (-(n+m)/(n-m)) * T(n-1,m) ,  0 < m < n
> > T(n,n) = ((4*n-2)/n) * T(n-1,n-1)
> >
> > A much cleaner and faster algorithm than what I was doing before!
> 
> Congratulations!
> 
> Orthogonal polynomials intrigue me because of their applications (I am
> interested in classical quantum mechanics, the Schrodinger equation and all
> that), but I have never completely understood them or how do generate them with
> recursion relations and generating functions. I eyeballed your question for a
> time trying to see the pattern, and I am glad you found it.
> 
> Orthogonal polynomials are often associated with particular ODEs (I am thinking
> here of Legendre, Laguerre, and Hermite polynomials). Any idea which ODE, if
> any, your equations are associated with?

Notice that legendre Pol are defined in [-1,1] . This interval can also be 
described as sin(theta) , theta from south-pole to north-pole on a sphere. 

These polynomials are a subset of Spherical Harmonics corresponding to 
wavenumber zero round equator.

The general spherical harmonics are :

           Y_m,n (theta,lambda) = Pol_m,n (sin(theta)) exp(i m lambda) .

The wave 0 Legendre Pol is:

           P(x) = Pol_0,n (x) , x in [-1,1] , x=sin(theta) is redundant.


Y_m,n are eigenfunctions of the Laplacian on a sphere:

           Del^2  Y_m,n ()  = n(n+1) Y_m,n ()


Johannes