\\% Chebyshev polynomials.
\\ Author: Joerg Arndt
\\ License: GPL version 3 or later
\\ online at http://www.jjj.de/pari/
\\ version: 2011-January-19 (12:52)

chebyT(n,x)= return( ([1, x]*[0,-1; 1,2*x]^n)[1] );
chebyU(n,x)= return( ([1, 2*x]*[0,-1; 1,2*x]^n)[1] );


chebyTsym(n, x)=
{
    local(b, s);
    if ( n<0,  n = -n );  \\ symmetry
    if ( n==1, return( x ) );  \\ avoid division by zero
    b = 2^(n-1);
    if ( 0==n%2, if ( 0==n%4, s=+1, s=-1 ), s=0 );
    forstep (k=n, 1, -2,
        s += b*x^(k);
        b *= -(k*(k-1))/((n+k-2)*(n-k+2));
    );
    return( s );
} /* ----- */


chebyUsym(n, x)=
{
    local(b, s);
    if ( n<=0,
         if ( n>=-2, return( n+1 ) );
         return( -chebyUsym( -n-2, x ) );  \\ use symmetry
    );
    n += 1;
    b = 2^(n-1) / n;
    s = 0;
    forstep (k=n, 1, -2,
        s += (k)*b*x^(k-1);
        b *= -(k*(k-1))/((n+k-2)*(n-k+2));
    );
    return( s );
} /* ----- */

\\ ==== end of file ====