Singular Manual: chebyshev

| #### D.4.6.4 chebyshev Procedure from library `decomp.lib` (see decomp_lib). 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. Example:
    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

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