O16-C056
> restart:
> ps:=p->sum(subs(t=cos((2*'k'-1)*Pi/4),p),'k'=1..4)*Pi/4;
               /   4                                     \   
               |-----                                    |   
               | \                                       |   
             1 |  )        /       /1               \   \|   
  ps := p -> - | /     subs|t = cos|- (2 'k' - 1) Pi|, p|| Pi
             4 |-----      \       \4               /   /|   
               \'k' = 1                                  /   
> b:=[seq((t^i),i=0..7)] ;
                   [       2   3   4   5   6   7]
              b := [1, t, t , t , t , t , t , t ]
> b[4];
                                3
                               t 
> c:=[seq(ps(b[i]),i=1..8)];
                    [       Pi     Pi     Pi   ]
               c := [Pi, 0, --, 0, --, 0, --, 0]
                    [       2      4      8    ]
> ph:=p->int(p/sqrt(1-t^2),t=-1..1);
                        /     p                     \
          ph := p -> int|------------, t = (-1 .. 1)|
                        |    /     2\               |
                        \sqrt\1 - t /               /
> d:=[seq(ph(b[i]),i=1..8)];
                  [       Pi     3 Pi     5 Pi   ]
             d := [Pi, 0, --, 0, ----, 0, ----, 0]
                  [       2       8        16    ]
> q[0]:=1;q[1]:=t;for k from 2 to 8 do q[k]:=expand(2*t*q[k-1]-q[k-2]) od:
                           q[0] := 1
                           q[1] := t
> seq(ps(q[l])-ph(q[l]),l=0..7);
                    0, 0, 0, 0, -Pi, 0, 0, 0
> ps(4,t[7])-ph(t[7]);
> #ph(p)=ps(p) si deg(p)<=n-1
                               0