jacobi_p expansion? , generalizing the thought

Should jacobi_p be expanded into a polynomial?
Tradition suggests that if there is a justification for
the form, e.g.  we can differentiate jacobi_p and get
the answer faster and neater in terms of jacobi_p,  then
it should not be >>automatically<< expanded. 

Like  (x+1)^10.  Should it be expanded? Sometimes: we have an expand (%);
So we could have a jacobi_p_expand.
or like expop, which automatically sets an expansion limit, we
could have a jacobi_p_expand  flag to the simplifier.

This is really a specific case of a pervasive issue.

Should complex symbolism be retained when it can be
re-expressed in other forms.  Do we need  difference, as in x-y?
not really.  It's  x+ (-1)*y   so all we need is multiplication by
negative constants.  x/y is  x*y^(-1), so we don't need
quotient, it seems. The Macsyma simplifier DOES do these!  BUT..
We tend to keep tan(x) around even though it is sin/cos.  or it could
be expressed in complex exponentials. 

This is a swamp.  You could say "simplify with respect to preferences"
and then list the functions you'd like to see. Mathematica and Maple
have tried this.  It's still a swamp.

Perhaps one way to handle this is to have a (huge?) list of rules,
and have the user select those that should be applied during simplification.
The possibility then exists of loops,  like sin/cos --> 
complex_exponential --> trig.
Maybe we could compute which selections cause loops and have "radio buttons"
where you can only choose one of a set of "reversal" transforms.

Sometimes you need to "hill climb"..  First combine all expressions by
expansions [see if you get something simpler] and then reexpress by some
other transformations.

Or you could say "make it small" and try to let the computer figure
out how.  I think this is a bad idea except for very very small inputs.
Mathematica has such a program. Maybe called FullSimplify ?


There are, incidentally, some papers on Chebyshev polynomials that I wrote
with Ted Einwohner, quite a while ago. There may be some suggestions about
dealing with this expansion/ non-expansion issue.
See www.cs.berkeley.edu/~fateman/papers/cheby-89.pdf



RJF




Barton Willis wrote:

>The Jacobi polynomials jacobi_p(n,a,b,x) are defined for all real a & b,
>but their weight function (1-x)^a (1+x)^b isn't integrable  for a,b in 
>(-inf, -1]. 
>Maybe for this reason,  for a, b  in (-inf,-1], specfun doesn't expand 
>jacobi_p(n,a,b,x) into a polynomial.  But orthopoly will
>
>(%i1) load("l:/orthopoly-0.94/orthopoly.lisp");
>
>(%i2)  hgfred([-2,2+1/2],[3/2],x);
>(%o2) 8*jacobi_p(2,1/2,-1,1-2*x)/15
>(%i3) ev(%);
>(%o3) 7*x^2/3-10*x/3+1
>
>Notes: 
>
>(1) orthopoly isn't in the maxima distribution; get it from
>
>http://www.unk.edu/acad/math/people/willisb/
>
>(2)  In the past few days, Ray fixed some things in hyp.lisp. 
>You'll need a very new CVS maxima to get the result in %o2 .
>
>(3) If I weren't so slothful, I'd change orthopoly to make the functions 
>work by simplification ---- then the ev in %i3 wouldn't be needed.
>
>Barton
>
>
>
>
>
>Raymond Toy <raymond.toy@ericsson.com>
>Sent by: maxima-admin@math.utexas.edu
>12/02/2004 04:04 PM
>
> 
>        To:     Maxima List <maxima@math.utexas.edu>
>        cc: 
>        Subject:        [Maxima] jacobi_p expansion?
>
>
>
>jacobi_p(2,1/2,-1,1-2*x) is a polynomial (of course).  But maxima
>doesn't expand that out.  Should it?
>
>Note that if n is an integer, jacobi_p(n,1/2,-1,1-2*x) gives a series
>expression, which I can then use via ev(%,n=2) to get a polynomial.
>
>Ray
>
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