Singular

D.4.6.4 chebyshev

Usage:

chebyshev(n); n int, n >= 0
chebyshev(n,c); n int, n >= 0, c number, c!=0

Return:

poly, the [monic] nth Chebyshev polynomial of the first kind.
The polynomials are defined in the first variable, say x, of the basering.

Note:

The (generalized) Chebyshev polynomials of the first kind can be defined by the recursion: .

These polynomials commute by composition:

.
For c=1, we obtain the standard (non monic) Chebyshev polynomials

which satisfy .
For c=2 (default), we obtain the monic Chebyshev polynomials

which satisfy the relation .
By default the monic Chebyshev polynomials are returned:

chebyshev(n) and chebyshev(n,1).
It holds and more generally

That is subst(chebyshev(n,c),var(1),c*var(1))= c*chebyshev(n,1).

If char(basering) = 2, then

, and so on.

LIB "decomp.lib";
ring r = 0,x,lp;
// The monic Chebyshev polynomials
chebyshev(0);
==> 2
chebyshev(1);
==> x
chebyshev(2);
==> x2-2
chebyshev(3);
==> x3-3x
// These polynomials commute
compose(chebyshev(2),chebyshev(6)) ==
compose(chebyshev(6),chebyshev(2));
==> 1
// The standard Chebyshev polynomials
chebyshev(0,1);
==> 1
chebyshev(1,1);
==> x
chebyshev(2,1);
==> 2x2-1
chebyshev(3,1);
==> 4x3-3x
// -----------------------------------------------------------------------
// The relation for the various Chebyshev polynomials
5*chebyshev(3,1)==subst(chebyshev(3,5),x,5x);
==> 1
// -----------------------------------------------------------------------
// char 2 case
ring r2 = 2,x,dp;
chebyshev(2);
==> 1
chebyshev(3);
==> x