Updates:
Cc: ppue...@googlemail.com
Comment #6 on issue 4133 by asmeu...@gmail.com: Suggestion: implement
classical orthogonal polynomials
http://code.google.com/p/sympy/issues/detail?id=4133
Greg, if you want to take a shot at implementing some of these things
yourself, go for it. Raoul and others can point you in the right direction.
I guess what we need is a orthopolysimp, which handles all these
identities. That or try to integrate them all into the expand/rewrite
frameworks. One issue is that it's not clear which way you would want to
go. chebyshevt(2,x)**2 seems simpler to me, but to you
chebyshevt(4,x)/2+chebyshevt(0,x)/2 is simpler.
The first step would be to remove the automatic evaluation of explicit
degrees into polynomials, so that such representations are possible. We
have .rewrite('polynomial') and chebyshevt_poly for those.
On a related note, we should maybe add some functions to efficiently
rewrite a given function or polynomial as a linear combination of
orthogonal polynomials. Sure, you could do the integration, but there may
be more efficient ways for some functions, and also it should be doable
even without remembering the exact integral to use. I guess this would be
part of a bigger Fourier series framework.
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