> restart:
> with(PolynomialTools):
> with(Statistics):
> w:=PDF(RandomVariable(Uniform(-1,1)),x) assuming x>0, x<1;
> assume(n,integer,n>1,alpha,real,alpha>0);
> # Legendre
> R:=n*Pn=(2*n-1)*xPn_1+0*Pn_1-(n-1)*Pn_2;
> normF:=sqrt(1/(2*n+1));
> S:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-1,normF
> )*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R);
> map(_x->factor(simplify(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn
> _1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),
> Qn_2)/coeff(lhs(S),Qn)]);
> C0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=0,1/normF*Vect
> or([1])));
> C1:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=1,1/normF*Vect
> or([0,1])));
> U:=1:V:=x:
> for i from 2 to 5 do
>  
> W:=map(factor,collect(subs(n=i,xPn_1=x*V,Pn_1=V,Pn_2=U,solve(R,Pn)),x)
> ):
>  
> print(map(_x->factor(simplify(_x,symbolic)),W),"->",map(_x->factor(sim
> plify(_x,symbolic)),collect(subs(n=i,W/normF),x)));
>   U:=V:
>   V:=W:
> od:
> U:=C0[1]:V:=C1[1]+C1[2]*x:
> evalf(CoefficientList(U,x));
> evalf(CoefficientList(V,x));
> for i from 2 to 9 do
>  
> W:=map(factor,collect(subs(n=i,xQn_1=x*V,Qn_1=V,Qn_2=U,solve(S,Qn)),x)
> ):
>   print(evalf(Int(W*W*w,x=-1..1)));
>   print(evalf(CoefficientList(W,x)));
>   U:=V:
>   V:=W:
> od:

                               w := 1/2


            R := n~ Pn = (2 n~ - 1) xPn_1 - (n~ - 1) Pn_2


                                       1
                        normF := -------------
                                           1/2
                                 (2 n~ + 1)


                n~ Qn                 1/2         (n~ - 1) Qn_2
       S := ------------- = (2 n~ - 1)    xQn_1 - -------------
                      1/2                                   1/2
            (2 n~ + 1)                            (2 n~ - 3)


                 1/2           1/2                          1/2
       (2 n~ - 1)    (2 n~ + 1)          (n~ - 1) (2 n~ + 1)
      [---------------------------, 0, - ----------------------]
                   n~                                 1/2
                                            (2 n~ - 3)    n~


                              C0 := [1]


                                   [ 0  ]
                             C1 := [    ]
                                   [ 1/2]
                                   [3   ]


                     2                 1/2  2    1/2
                  3 x               3 5    x    5
                  ---- - 1/2, "->", --------- - ----
                   2                    2        2


                 3                     1/2  3        1/2
            5/2 x  - 3/2 x, "->", 5/2 7    x  - 3/2 7    x


                   4         2               4         2
       3/8 + 35/8 x  - 15/4 x , "->", 105/8 x  - 45/4 x  + 9/8


        5         3
  63/8 x  - 35/4 x  + 15/8 x, "->",

               1/2  5          1/2  3          1/2
        63/8 11    x  - 35/4 11    x  + 15/8 11    x


                                 [1.]


                          [0., 1.732050808]


                             1.000000000


                   [-1.118033988, 0., 3.354101966]


                             1.000000000


                 [0., -3.968626966, 0., 6.614378278]


                             1.000000000


           [1.125000000, 0., -11.25000000, 0., 13.12500000]


                             1.000000000


         [0., 6.218671481, 0., -29.02046691, 0., 26.11842022]


                             1.000000000


  [-1.126734773, 0., 23.66143024, 0., -70.98429073, 0., 52.05514653]


                             1.000000000


  [0., -8.472151069, 0., 76.24935962, 0., -167.7485912, 0.,

        103.8443660]


                             1.000000000


  [1.127411695, 0., -40.58682101, 0., 223.2275155, 0., -386.9276936,

        0., 207.2826930]


                             1.000000000


  [0., 10.72697787, 0., -157.3290088, 0., 613.5831342, 0.,

        -876.5473345, 0., 413.9251302]

> restart:
> with(PolynomialTools):
> with(Statistics):
> w:=PDF(RandomVariable(Normal(0,1)),x);
> # Hermite
> R:=Pn=xPn_1-(n-1)*Pn_2;
> normF:=sqrt(GAMMA(n+1));
> S:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-1,normF
> )*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R);
> map(_x->factor(simplify(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn
> _1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),
> Qn_2)/coeff(lhs(S),Qn)]);
> C0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=0,1/normF*Vect
> or([1])));
> C1:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=1,1/normF*Vect
> or([0,1])));
> U:=1:V:=x:
> for i from 2 to 5 do
>  
> W:=map(factor,collect(subs(n=i,xPn_1=x*V,Pn_1=V,Pn_2=U,solve(R,Pn)),x)
> ):
>  
> print(map(_x->factor(simplify(_x,symbolic)),W),"->",map(_x->factor(sim
> plify(_x,symbolic)),collect(subs(n=i,W/normF),x)));
>   U:=V:
>   V:=W:
> od:
> U:=C0[1]:V:=C1[1]+C1[2]*x:
> evalf(CoefficientList(U,x));
> evalf(CoefficientList(V,x));
> for i from 2 to 9 do
>  
> W:=map(factor,collect(subs(n=i,xQn_1=x*V,Qn_1=V,Qn_2=U,solve(S,Qn)),x)
> ):
>   print(evalf(Int(W*W*w,x=-infinity..infinity)));
>   print(evalf(CoefficientList(W,x)));
>   U:=V:
>   V:=W:
> od:

                                            2
                                1/2        x
                               2    exp(- ----)
                                           2
                      w := 1/2 ----------------
                                      1/2
                                    Pi


                    R := Pn = xPn_1 - (n - 1) Pn_2


                                            1/2
                       normF := GAMMA(n + 1)


                   1/2
  S := GAMMA(n + 1)    Qn =

                1/2                             1/2
        GAMMA(n)    xQn_1 - (n - 1) GAMMA(n - 1)    Qn_2


                                          1/2
                         1         (n - 1)
                       [----, 0, - ----------]
                         1/2           1/2
                        n             n


                              C0 := [1]


                                    [0]
                              C1 := [ ]
                                    [1]


                                    1/2  2    1/2
                      2            2    x    2
                     x  - 1, "->", ------- - ----
                                      2       2


                3                   1/2  3        1/2
               x  - 3 x, "->", 1/6 6    x  - 1/2 6    x


           4      2              1/2  4        1/2  2        1/2
      3 + x  - 6 x , "->", 1/12 6    x  - 1/2 6    x  + 1/4 6


  5       3                      1/2  5         1/2  3         1/2
 x  - 10 x  + 15 x, "->", 1/60 30    x  - 1/6 30    x  + 1/4 30    x


                                 [1.]


                               [0., 1.]


                             1.000000000


                  [-0.7071067810, 0., 0.7071067810]


                             1.000000000


                 [0., -1.224744872, 0., 0.4082482906]


                             1.000000000


          [0.6123724358, 0., -1.224744872, 0., 0.2041241453]


                             1.000000000


       [0., 1.369306394, 0., -0.9128709291, 0., 0.09128709291]


                             1.000000000


  [-0.5590169943, 0., 1.677050983, 0., -0.5590169943, 0.,

        0.03726779963]


                             1.000000000


  [0., -1.479019946, 0., 1.479019946, 0., -0.2958039892, 0.,

        0.01408590425]


                             1.000000000


  [0.5229125167, 0., -2.091650067, 0., 1.045825033, 0., -0.1394433378,

        0., 0.004980119206]


                             1.000000000


  [0., 1.568737550, 0., -2.091650066, 0., 0.6274950201, 0.,

        -0.05976143047, 0., 0.001660039735]

> restart:
> with(PolynomialTools):
> with(Statistics):
> w:=PDF(RandomVariable(Gamma(1,alpha+1)),x);
> # Laguerre
> R:=n*Pn=-xPn_1+(2*n-1+alpha)*Pn_1-(n-1+alpha)*Pn_2;
> normF:=sqrt(GAMMA(n+alpha+1)/GAMMA(alpha+1)/GAMMA(n+1));
> S:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-1,normF
> )*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R);
> map(_x->factor(simplify(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn
> _1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),
> Qn_2)/coeff(lhs(S),Qn)]);
> C0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=0,1/normF*Vect
> or([1])));
> C1:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=1,1/normF*Vect
> or([1+alpha,-1])));
> alpha_num:=2.5:
> U:=1:V:=1+alpha-x:
> for i from 2 to 3 do
>  
> W:=map(_x->factor(simplify(_x,symbolic)),collect(subs(n=i,xPn_1=x*V,Pn
> _1=V,Pn_2=U,solve(R,Pn)),x)):
>  
> print(map(_x->factor(simplify(_x,symbolic)),W),"->",map(_x->factor(sim
> plify(_x,symbolic)),collect(subs(n=i,W/normF),x)),collect(evalf(subs(n
> =i,alpha=alpha_num,W/normF)),x));
>   U:=V:
>   V:=W:
> od:
> U:=C0[1]:V:=C1[1]+C1[2]*x:
> evalf(CoefficientList(subs(alpha=alpha_num,U),x));
> evalf(CoefficientList(subs(alpha=alpha_num,V),x));
> for i from 2 to 9 do
>  
> W:=map(_x->factor(simplify(_x,symbolic)),collect(subs(n=i,xQn_1=x*V,Qn
> _1=V,Qn_2=U,alpha=alpha_num,solve(S,Qn)),x)):
>   print(evalf(Int(subs(alpha=alpha_num,W*W*w),x=0..infinity)));
>   print((CoefficientList(W,x)));
>   U:=V:
>   V:=W:
> od:

                    {        0                  x < 0
                    {
               w := {   alpha
                    {  x      exp(-x)
                    { ----------------        otherwise
                    { GAMMA(alpha + 1)


  R := n Pn = -xPn_1 + (2 n - 1 + alpha) Pn_1 - (n - 1 + alpha) Pn_2


                      /    GAMMA(n + alpha + 1)     \1/2
             normF := |-----------------------------|
                      \GAMMA(alpha + 1) GAMMA(n + 1)/


         /    GAMMA(n + alpha + 1)     \1/2
  S := n |-----------------------------|    Qn =
         \GAMMA(alpha + 1) GAMMA(n + 1)/

         /    GAMMA(n + alpha)     \1/2
        -|-------------------------|    xQn_1
         \GAMMA(alpha + 1) GAMMA(n)/

                             /    GAMMA(n + alpha)     \1/2
         + (2 n - 1 + alpha) |-------------------------|    Qn_1
                             \GAMMA(alpha + 1) GAMMA(n)/

                           /    GAMMA(n - 1 + alpha)     \1/2
         - (n - 1 + alpha) |-----------------------------|    Qn_2
                           \GAMMA(alpha + 1) GAMMA(n - 1)/


              1             2 n - 1 + alpha
  [- -------------------, -------------------,
                1/2  1/2             1/2  1/2
     (n + alpha)    n     (n + alpha)    n

                 1/2                1/2
          (n - 1)    (n - 1 + alpha)
        - -----------------------------]
                          1/2  1/2
               (n + alpha)    n


                              C0 := [1]


                             [            1/2 ]
                             [ (alpha + 1)    ]
                             [                ]
                       C1 := [        1       ]
                             [- --------------]
                             [             1/2]
                             [  (alpha + 1)   ]


    2
   x                     (alpha + 1) alpha
  ---- - (2 + alpha) x + ----------------- + 1 + alpha, "->",
   2                             2

                     1/2  2                1/2            1/2
                    2    x                2    (2 + alpha)    x
        ------------------------------- - ---------------------
                     1/2            1/2                 1/2
        2 (alpha + 1)    (2 + alpha)         (alpha + 1)

                      1/2            1/2  1/2
           (alpha + 1)    (2 + alpha)    2
         + ----------------------------------,
                           2

                      2
        0.1781741613 x  - 1.603567452 x + 2.806243040


      3                 2
     x     (3 + alpha) x    (3 + alpha) (2 + alpha) x
  - ---- + -------------- - -------------------------
     6           2                      2

           (alpha + 1) (2 + alpha) alpha   (alpha + 1) alpha
         + ----------------------------- + ----------------- + 1
                         6                         2

         + alpha, "->",

                              1/2  3
                             6    x
        - ----------------------------------------------
                       1/2            1/2            1/2
          6 (alpha + 1)    (2 + alpha)    (3 + alpha)

                          1/2  1/2  2
               (3 + alpha)    6    x
         + -------------------------------
                        1/2            1/2
           2 (alpha + 1)    (2 + alpha)

                      1/2            1/2  1/2
           (3 + alpha)    (2 + alpha)    6    x
         - ------------------------------------
                                  1/2
                     2 (alpha + 1)

                      1/2            1/2            1/2  1/2
           (alpha + 1)    (2 + alpha)    (3 + alpha)    6
         + -------------------------------------------------,
                                   6

                        3                 2
        -0.04386344634 x  + 0.7237468645 x  - 3.256860890 x

         + 3.799671039


                                 [1.]


                     [1.870828693, -0.5345224839]


                             1.000000001


              [2.806243040, -1.603567452, 0.1781741614]


                             1.000000002


      [3.799671039, -3.256860890, 0.7237468648, -0.04386344637]


                             1.000000001


  [4.843649192, -5.535599076, 1.845199693, -0.2236605687,

        0.008602329571]


                             1.000000001


  [5.932234508, -8.474620723, 3.766498103, -0.6848178366,

        0.05267829511, -0.001404754537]


                             1.000000002


  [7.060771360, -12.10417948, 6.724544156, -1.630192524, 0.1880991373,

        -0.01003195398, 0.0001967049802]


                             1.000000003


  [8.225549714, -16.45109943, 10.96739962, -3.323454434, 0.5113006820,

        -0.04090405454, 0.001604080569, -0.00002412151234]


                             1.000000004


  [9.423551064, -21.53954527, 16.75297966, -6.091992609, 1.171537040,

        -0.1249639508, 0.007350820636, -0.0002210773124,

                       -5
        0.2631872767 10  ]


                             1.000000003


  [10.65227215, -27.39155695, 24.34805062, -10.32947604, 2.383725238,

        -0.3178300313, 0.02492784559, -0.001124564463,

                                          -6
        0.00002677534436, -0.2586989793 10  ]

> restart:
> with(PolynomialTools):
> with(Statistics):
> w:=simplify(PDF(-1+2*RandomVariable(Beta(b+1,a+1)),x),symbolic);
> # Jacobi
> R:=Pn=(2*n+a+b)*(2*n-1+a+b)/2/(n+a+b)/n*xPn_1+(a-b)*(a+b)*(2*n-1+a+b)/
> 2/(n+a+b)/(2*n-2+a+b)/n*Pn_1-(a+n-1)*(b+n-1)*(2*n+a+b)/(2*n-2+a+b)/(n+
> a+b)/n*Pn_2;
> normF:=convert(sqrt(2^(a+b+1)*GAMMA(n+a+1)*GAMMA(n+b+1)/(2*n+a+b+1)/GA
> MMA(n+a+b+1)/GAMMA(n+1)/2^(a+b+1)/Beta(b+1,a+1)),GAMMA);
> S:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-1,normF
> )*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R);
> map(_x->factor(simplify(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn
> _1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),
> Qn_2)/coeff(lhs(S),Qn)]);
> C0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=0,1/normF*Vect
> or(CoefficientList(simplify(JacobiP(0,a,b,x)),x))));
> C1:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=1,1/normF*Vect
> or(CoefficientList(simplify(JacobiP(1,a,b,x)),x))));
> a_num:=2.5:b_num:=3.5:
> U:=1:V:=simplify(JacobiP(1,a,b,x)):
> for i from 2 to 5 do
>  
> W:=map(_x->factor(simplify(_x,symbolic)),collect(subs(n=i,xPn_1=x*V,Pn
> _1=V,Pn_2=U,solve(R,Pn)),x)):
>  
> print(subs(u=1-x,map(_x->factor(simplify(_x,symbolic)),collect(subs(x=
> 1-u,W),u))),"->",map(_x->factor(simplify(_x,symbolic)),collect(subs(n=
> i,W/normF),x)),collect(evalf(subs(n=i,a=a_num,b=b_num,W/normF)),x));
>   U:=V:
>   V:=W:
> od:
> U:=C0[1]:V:=C1[1]+C1[2]*x:
> evalf(CoefficientList(subs(a=a_num,b=b_num,U),x));
> evalf(CoefficientList(subs(a=a_num,b=b_num,V),x));
> for i from 2 to 10 do
>  
> W:=map(_x->factor(simplify(_x,symbolic)),collect(subs(n=i,xQn_1=x*V,Qn
> _1=V,Qn_2=U,a=a_num,b=b_num,solve(S,Qn)),x)):
>   print(evalf(Int(subs(a=a_num,b=b_num,W*W*w),x=-1..1)));
>   print((CoefficientList(W,x)));
>   U:=V:
>   V:=W:
> od:

               {               0                      x < -1
               {
               {                b            a
          w := {     (x/2 + 1/2)  (1/2 - x/2)
               { 1/2 -------------------------        x < 1
               {        Beta(b + 1, a + 1)
               {
               {               0                      1 <= x


            (2 n + a + b) (2 n - 1 + a + b) xPn_1
  R := Pn = -------------------------------------
                       2 (n + a + b) n

           (a - b) (b + a) (2 n - 1 + a + b) Pn_1
         + --------------------------------------
             2 (n + a + b) (2 n - 2 + a + b) n

           (a + n - 1) (b + n - 1) (2 n + a + b) Pn_2
         - ------------------------------------------
                (2 n - 2 + a + b) (n + a + b) n


  normF := (GAMMA(n + a + 1) GAMMA(n + b + 1) GAMMA(b + 2 + a)/(

        (2 n + a + b + 1) GAMMA(n + a + b + 1) GAMMA(n + 1)

                                   1/2
        GAMMA(b + 1) GAMMA(a + 1)))


  S := (GAMMA(n + a + 1) GAMMA(n + b + 1) GAMMA(b + 2 + a)/(

        (2 n + a + b + 1) GAMMA(n + a + b + 1) GAMMA(n + 1)

                                   1/2
        GAMMA(b + 1) GAMMA(a + 1)))    Qn = 1/2 (2 n + a + b)

        (2 n - 1 + a + b) (GAMMA(n + a) GAMMA(n + b) GAMMA(b + 2 + a)

        /((2 n - 1 + a + b) GAMMA(n + a + b) GAMMA(n) GAMMA(b + 1)

                      1/2
        GAMMA(a + 1)))    xQn_1/((n + a + b) n) + 1/2 (a - b) (b + a)

        (2 n - 1 + a + b) (GAMMA(n + a) GAMMA(n + b) GAMMA(b + 2 + a)

        /((2 n - 1 + a + b) GAMMA(n + a + b) GAMMA(n) GAMMA(b + 1)

                      1/2
        GAMMA(a + 1)))    Qn_1/((n + a + b) (2 n - 2 + a + b) n) -

        (a + n - 1) (b + n - 1) (2 n + a + b) (GAMMA(a + n - 1)

        GAMMA(b + n - 1) GAMMA(b + 2 + a)/((2 n - 3 + a + b)

        GAMMA(n - 1 + a + b) GAMMA(n - 1) GAMMA(b + 1) GAMMA(a + 1)))^

        1/2
            Qn_2/((2 n - 2 + a + b) (n + a + b) n)


                    1/2                                1/2
   (2 n - 1 + a + b)    (2 n + a + b) (2 n + a + b + 1)
  [-------------------------------------------------------,
                  1/2        1/2            1/2  1/2
         2 (n + a)    (n + b)    (n + a + b)    n

                         1/2                                  1/2
        (2 n - 1 + a + b)    (a - b) (b + a) (2 n + a + b + 1)

           /           1/2        1/2            1/2  1/2
          /  (2 (n + a)    (n + b)    (n + a + b)    n
         /

                                                       1/2
        (2 n - 2 + a + b)), - (2 n + a + b) (a + n - 1)

                   1/2                1/2        1/2
        (b + n - 1)    (n - 1 + a + b)    (n - 1)

                         1/2   /         1/2        1/2
        (2 n + a + b + 1)     /  ((n + a)    (n + b)
                             /

                         1/2            1/2  1/2
        (2 n - 3 + a + b)    (n + a + b)    n    (2 n - 2 + a + b))]


                              C0 := [1]


                        [                    1/2   ]
                        [ (a - b) (b + 3 + a)      ]
                        [ -----------------------  ]
                        [          1/2        1/2  ]
                        [ 2 (b + 1)    (a + 1)     ]
                  C1 := [                          ]
                        [                       1/2]
                        [(b + 2 + a) (b + 3 + a)   ]
                        [--------------------------]
                        [          1/2        1/2  ]
                        [ 2 (b + 1)    (a + 1)     ]


                                 2
  (a + 4 + b) (b + 3 + a) (1 - x)    (b + 3 + a) (2 + a) (1 - x)
  -------------------------------- - ---------------------------
                 8                                2

                                       2
           (a + 4 + b) (b + 3 + a)    a           (a - b) (b + 3 + a)
         + ----------------------- + ---- - a/8 + -------------------
                      8               8                    4

                         2
                        b     a b                                 1/2
         - 1/2 - b/8 + ---- - ---, "->", (a + 4 + b) (b + 3 + a) 2
                        8      4

                   1/2            1/2  2   /           1/2        1/2
        (5 + a + b)    (b + 2 + a)    x   /  (8 (b + 1)    (b + 2)
                                         /

               1/2        1/2
        (a + 1)    (2 + a)   )

                                1/2            1/2            1/2
           (a - b) (b + 3 + a) 2    (5 + a + b)    (b + 2 + a)    x
         + --------------------------------------------------------
                         1/2        1/2        1/2        1/2
                4 (b + 1)    (b + 2)    (a + 1)    (2 + a)

             2                        2   1/2            1/2
         + (a  - a - 2 a b - 4 - b + b ) 2    (5 + a + b)

                   1/2   /           1/2        1/2        1/2
        (b + 2 + a)     /  (8 (b + 1)    (b + 2)    (a + 1)
                       /

               1/2                2
        (2 + a)   ), 7.559289462 x  - 1.511857892 x - 0.7559289462


                                               3
    (6 + a + b) (5 + a + b) (a + 4 + b) (1 - x)
  - --------------------------------------------
                         48

                                                  2
           (5 + a + b) (a + 4 + b) (3 + a) (1 - x)
         + ----------------------------------------
                              8

           (a + 4 + b) (3 + a) (2 + a) (1 - x)
         - -----------------------------------
                            4

           (6 + a + b) (5 + a + b) (a + 4 + b)
         + -----------------------------------
                           48

                     2                       2
           (a - b) (a  - 3 a - 2 a b - 16 + b  - 3 b)
         + ------------------------------------------
                               48

                         2                2
           (a + 4 + b) (a  - a - 2 a b + b  - 6 - b)
         + -----------------------------------------
                              16

           (a - b) (5 + a + b) (a + 4 + b)
         + -------------------------------, "->", (6 + a + b)
                         16

                                 1/2            1/2            1/2
        (5 + a + b) (a + 4 + b) 6    (7 + a + b)    (b + 2 + a)

                   1/2  3   /            1/2        1/2        1/2
        (b + 3 + a)    x   /  (48 (b + 1)    (b + 2)    (b + 3)
                          /

               1/2        1/2        1/2
        (a + 1)    (2 + a)    (3 + a)   ) + (a - b) (5 + a + b)

                     1/2            1/2            1/2            1/2
        (a + 4 + b) 6    (7 + a + b)    (b + 2 + a)    (b + 3 + a)

         2   /            1/2        1/2        1/2        1/2
        x   /  (16 (b + 1)    (b + 2)    (b + 3)    (a + 1)
           /

               1/2        1/2
        (2 + a)    (3 + a)   ) + (a + 4 + b)

          2                2           1/2            1/2
        (a  - a - 2 a b + b  - 6 - b) 6    (7 + a + b)

                   1/2            1/2     /            1/2        1/2
        (b + 2 + a)    (b + 3 + a)    x  /  (16 (b + 1)    (b + 2)
                                        /

               1/2        1/2        1/2        1/2
        (b + 3)    (a + 1)    (2 + a)    (3 + a)   ) + (a - b)

          2                       2         1/2            1/2
        (a  - 3 a - 2 a b - 16 + b  - 3 b) 6    (7 + a + b)

                   1/2            1/2   /            1/2        1/2
        (b + 2 + a)    (b + 3 + a)     /  (48 (b + 1)    (b + 2)
                                      /

               1/2        1/2        1/2        1/2                3
        (b + 3)    (a + 1)    (2 + a)    (3 + a)   ), 17.45743122 x

                        2
         - 4.364357804 x  - 4.364357804 x + 0.4364357804


                                                         4
  (5 + a + b) (6 + a + b) (8 + a + b) (7 + a + b) (1 - x)
  --------------------------------------------------------
                            384

                                                              3
           (5 + a + b) (7 + a + b) (6 + a + b) (4 + a) (1 - x)
         - ----------------------------------------------------
                                    48

                                                          2
           (5 + a + b) (6 + a + b) (4 + a) (3 + a) (1 - x)
         + ------------------------------------------------
                                  16

           (5 + a + b) (4 + a) (3 + a) (2 + a) (1 - x)
         - -------------------------------------------
                               12

                                                              4
           (5 + a + b) (6 + a + b) (8 + a + b) (7 + a + b)   a
         + ----------------------------------------------- + ---
                                 384                         384

            3       3     2
           a  b    a     a  b
         - ---- - ---- + ----
            96     64     64

                                                             2      3
           (5 + a + b) (7 + a + b) (6 + a + b) (a - b)   37 a    a b
         + ------------------------------------------- - ----- - ----
                               96                         384     96

           7 a
         + ---
           64

                                     2                    2
           (5 + a + b) (6 + a + b) (a  - a - 2 a b - 8 + b  - b)
         + -----------------------------------------------------
                                    64

                      3
           43 a b    b
         + ------ - ----
            192      64

                                 2                       2
           (5 + a + b) (a - b) (a  - 3 a - 2 a b - 22 + b  - 3 b)
         + ------------------------------------------------------
                                     96

                  4              2    2  2      2
           7 b   b           37 b    a  b    a b
         + --- + --- + 3/8 - ----- + ----- + ----, "->", (5 + a + b)
           64    384          384     64      64

                                             1/2            1/2
        (6 + a + b) (8 + a + b) (7 + a + b) 6    (9 + a + b)

                   1/2            1/2            1/2  4   /
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    x   /  (192
                                                        /

               1/2        1/2        1/2        1/2        1/2
        (b + 1)    (b + 2)    (b + 3)    (4 + b)    (a + 1)

               1/2        1/2        1/2
        (2 + a)    (3 + a)    (4 + a)   ) + (5 + a + b) (7 + a + b)

                             1/2            1/2            1/2
        (6 + a + b) (a - b) 6    (9 + a + b)    (b + 2 + a)

                   1/2            1/2  3   /            1/2
        (b + 3 + a)    (a + 4 + b)    x   /  (48 (b + 1)
                                         /

               1/2        1/2        1/2        1/2        1/2
        (b + 2)    (b + 3)    (4 + b)    (a + 1)    (2 + a)

               1/2        1/2
        (3 + a)    (4 + a)   ) + (5 + a + b) (6 + a + b)

          2                    2       1/2            1/2
        (a  - a - 2 a b - 8 + b  - b) 6    (9 + a + b)

                   1/2            1/2            1/2  2   /
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    x   /  (32
                                                        /

               1/2        1/2        1/2        1/2        1/2
        (b + 1)    (b + 2)    (b + 3)    (4 + b)    (a + 1)

               1/2        1/2        1/2
        (2 + a)    (3 + a)    (4 + a)   ) + (5 + a + b) (a - b)

          2                       2         1/2            1/2
        (a  - 3 a - 2 a b - 22 + b  - 3 b) 6    (9 + a + b)

                   1/2            1/2            1/2     /
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    x  /  (48
                                                       /

               1/2        1/2        1/2        1/2        1/2
        (b + 1)    (b + 2)    (b + 3)    (4 + b)    (a + 1)

               1/2        1/2        1/2      4      3        3
        (2 + a)    (3 + a)    (4 + a)   ) + (a  - 4 a  b - 6 a

              2        2  2       2        3               2
         + 6 a  b + 6 a  b  - 37 a  - 4 a b  + 42 a + 6 a b  + 86 a b

              3       2           4         1/2            1/2
         - 6 b  - 37 b  + 42 b + b  + 144) 6    (9 + a + b)

                   1/2            1/2            1/2   /
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)     /  (192
                                                     /

               1/2        1/2        1/2        1/2        1/2
        (b + 1)    (b + 2)    (b + 3)    (4 + b)    (a + 1)

               1/2        1/2        1/2                4
        (2 + a)    (3 + a)    (4 + a)   ), 38.64367133 x

                        3                2
         - 11.04104895 x  - 16.56157342 x  + 2.760262238 x

         + 0.6900655592


  - (6 + a + b) (8 + a + b) (7 + a + b) (a + 10 + b) (9 + a + b)

               5
        (1 - x) /3840 + (6 + a + b) (8 + a + b) (7 + a + b)

                                   4
        (9 + a + b) (a + 5) (1 - x) /384 - (6 + a + b) (8 + a + b)

                                           3
        (7 + a + b) (a + 5) (4 + a) (1 - x) /96

                                                                  2
           (7 + a + b) (6 + a + b) (a + 5) (4 + a) (3 + a) (1 - x)
         + --------------------------------------------------------
                                      48

           (6 + a + b) (a + 5) (4 + a) (3 + a) (2 + a) (1 - x)
         - --------------------------------------------------- +
                                   48

        (6 + a + b) (8 + a + b) (7 + a + b) (a + 10 + b) (9 + a + b)/

                             4      3        3      2         2
        3840 + (6 + a + b) (a  - 4 a  b - 6 a  + 6 a  b - 49 a

              2  2                  3        2           4      3
         + 6 a  b  + 110 a b - 4 a b  + 6 a b  + 54 a + b  - 6 b

               2
         - 49 b  + 54 b + 240)/768 + (a - b) (7 + a + b) (6 + a + b)

          2                  2                             4      3
        (a  - 3 a - 2 a b + b  - 3 b - 28)/384 + (a - b) (a  - 4 a  b

               3       2      2  2       2
         - 10 a  - 65 a  + 6 a  b  + 10 a  b + 250 a + 190 a b

                3         2           4       3       2
         - 4 a b  + 10 a b  + 1024 + b  - 10 b  - 65 b  + 250 b)/3840

           (6 + a + b) (8 + a + b) (7 + a + b) (9 + a + b) (a - b)
         + ------------------------------------------------------- +
                                     768

        (6 + a + b) (8 + a + b) (7 + a + b)

          2                2
        (a  - a - 2 a b + b  - 10 - b)/384, "->", (6 + a + b)

                                                           1/2
        (8 + a + b) (7 + a + b) (a + 10 + b) (9 + a + b) 30

                    1/2            1/2            1/2            1/2
        (11 + a + b)    (b + 2 + a)    (b + 3 + a)    (a + 4 + b)

                   1/2  5   /              1/2        1/2        1/2
        (5 + a + b)    x   /  (1920 (b + 1)    (b + 2)    (b + 3)
                          /

               1/2        1/2        1/2        1/2        1/2
        (4 + b)    (5 + b)    (a + 1)    (2 + a)    (3 + a)

               1/2        1/2
        (4 + a)    (a + 5)   ) + (6 + a + b) (8 + a + b) (7 + a + b)

                              1/2             1/2            1/2
        (9 + a + b) (a - b) 30    (11 + a + b)    (b + 2 + a)

                   1/2            1/2            1/2  4   /
        (b + 3 + a)    (a + 4 + b)    (5 + a + b)    x   /  (384
                                                        /

               1/2        1/2        1/2        1/2        1/2
        (b + 1)    (b + 2)    (b + 3)    (4 + b)    (5 + b)

               1/2        1/2        1/2        1/2        1/2
        (a + 1)    (2 + a)    (3 + a)    (4 + a)    (a + 5)   ) +

        (6 + a + b) (8 + a + b) (7 + a + b)

          2                2             1/2             1/2
        (a  - a - 2 a b + b  - 10 - b) 30    (11 + a + b)

                   1/2            1/2            1/2            1/2
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    (5 + a + b)

         3   /             1/2        1/2        1/2        1/2
        x   /  (192 (b + 1)    (b + 2)    (b + 3)    (4 + b)
           /

               1/2        1/2        1/2        1/2        1/2
        (5 + b)    (a + 1)    (2 + a)    (3 + a)    (4 + a)

               1/2
        (a + 5)   ) + (a - b) (7 + a + b) (6 + a + b)

          2                  2               1/2             1/2
        (a  - 3 a - 2 a b + b  - 3 b - 28) 30    (11 + a + b)

                   1/2            1/2            1/2            1/2
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    (5 + a + b)

         2   /             1/2        1/2        1/2        1/2
        x   /  (192 (b + 1)    (b + 2)    (b + 3)    (4 + b)
           /

               1/2        1/2        1/2        1/2        1/2
        (5 + b)    (a + 1)    (2 + a)    (3 + a)    (4 + a)

               1/2                  4      3        3      2
        (a + 5)   ) + (6 + a + b) (a  - 4 a  b - 6 a  + 6 a  b

               2      2  2                  3        2           4
         - 49 a  + 6 a  b  + 110 a b - 4 a b  + 6 a b  + 54 a + b

              3       2                 1/2             1/2
         - 6 b  - 49 b  + 54 b + 240) 30    (11 + a + b)

                   1/2            1/2            1/2            1/2
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    (5 + a + b)    x

           /             1/2        1/2        1/2        1/2
          /  (384 (b + 1)    (b + 2)    (b + 3)    (4 + b)
         /

               1/2        1/2        1/2        1/2        1/2
        (5 + b)    (a + 1)    (2 + a)    (3 + a)    (4 + a)

               1/2              4      3         3       2      2  2
        (a + 5)   ) + (a - b) (a  - 4 a  b - 10 a  - 65 a  + 6 a  b

               2                            3         2           4
         + 10 a  b + 250 a + 190 a b - 4 a b  + 10 a b  + 1024 + b

               3       2            1/2             1/2
         - 10 b  - 65 b  + 250 b) 30    (11 + a + b)

                   1/2            1/2            1/2            1/2
        (b + 2 + a)    (b + 3 + a)    (a + 4 + b)    (5 + a + b)

           /              1/2        1/2        1/2        1/2
          /  (1920 (b + 1)    (b + 2)    (b + 3)    (4 + b)
         /

               1/2        1/2        1/2        1/2        1/2
        (5 + b)    (a + 1)    (2 + a)    (3 + a)    (4 + a)

               1/2                5                4                3
        (a + 5)   ), 83.37138529 x  - 26.05355790 x  - 52.10711581 x

                        2
         + 11.16581053 x  + 5.582905266 x - 0.4652421053


                                 [1.]


                     [-0.3779644730, 3.023715785]


                             1.000000001


              [-0.7559289457, -1.511857892, 7.559289463]


                             1.000000000


       [0.4364357803, -4.364357804, -4.364357804, 17.45743122]


                             1.000000001


  [0.6900655594, 2.760262237, -16.56157342, -11.04104895, 38.64367134

        ]


                             1.000000001


  [-0.4652421054, 5.582905265, 11.16581053, -52.10711581,

        -26.05355790, 83.37138531]


                             0.9999999999


  [-0.6579516949, -3.947710168, 27.63397118, 36.84529491,

        -147.3811796, -58.95247187, 176.8574156]


                             1.000000001


  [0.4828045497, -6.759263698, -20.27779109, 108.1482191, 108.1482191,

        -389.3335887, -129.7778630, 370.7938943]


                             1.000000002


  [0.6386903852, 5.109523082, -40.87618466, -81.75236932, 367.8856618,

        294.3085295, -981.0284316, -280.2938378, 770.8080540]


                             1.000000002


  [-0.4947274452, 7.915639125, 31.66255650, -189.9753391,

        -284.9630085, 1139.852034, 759.9013560, -2388.261405,

        -597.0653515, 1592.174270]


                             1.000000003


  [-0.6257862198, -6.257862196, 56.32075978, 150.1886928,

        -750.9434636, -901.1321560, 3304.151239, 1888.086422,

        -5664.259267, -1258.724283, 3272.683133]

> 
