A check for U_{m,2,j} in (27.27) of page 311.

The version number of Mathematica used to evaluate this notebook:

We begin by modifying a portion of the computation on pages 169-170 for Section 16.3.

$VersionNumber

p  =  7;    Date[]

TrueQ[ $VersionNumber >= 3.0]

a[2][z_]:=b[2][z]-D[b[1][z], z]/2-(b[1][z])^2/4

w[0, 0][z_]=1;

w[0, 1][z_]=Binomial[m, 2]*b[1][z];

w[0, 2][z_]=( Binomial[m + 1, 3]*a[2][z]+((m - 1)/2)*D[w[0, 1][z], z]+((m - 1)/(2  m))*(w[0, 1][z])^2 );

w[j_,i_][z_]:=w[i, j][z] /; j > i

Do[SumOfPowers[n,j_]=Sum[k^n,{k, j + 1, m}],{n, 0, 2*p + 1}]

K[-1,j_]=0;

K[0,j_]=1;

summandK[i_,j_]:=Expand[
          D[K[i, j], z] - ((m - 1)*b[1][z]*K[i, j])/
              2 + (m - i - j + 1)*(1 - i - j)*a[2][z]*
              K[i - 1, j]]

Do[ K[i,j_]=Module[{t, k},t = Expand[summandK[i - 1, k]];Sum[Coefficient[t, k, n]*SumOfPowers[n, j],{n, 0, 2*i - 1}]],{i, 1, p}]

L[i_,j_]:=Sum[ K[i - p, p]*K[j - q, q]*w[p, q][z],{p, 0, i},{q, 0, j}]

M[l_,n_,h_,i_]:=Binomial[m - h, l - h]*(Product[(l - r),{r, 1, l - h}])*Binomial[m - i + h, n - i + h]*(Product[(n - s),{s, 1, n - i + h}])*L[h, i - h]

U[-1,j_]=0;

U[0,j_]=1;

summandForU[i_,j_]:=Expand[
          D[U[i, j], z] + (i + j - m)*b[1][z]*
              U[i, j] + (m - i - j + 1)*(1 - i - j)*b[2][z]*
              U[i - 1, j]]

Do[U[i,j_]=Module[{u, r},u = Expand[summandForU[i - 1, r]];Sum[Coefficient[u, r, n]*SumOfPowers[n, j],{n, 0, 2*i - 1}]],{i, 1, p}]

b[1][z_]  =  x[1][z]/Binomial[m, 2]

b[2][
        z_]  =  (x[2][
              z]  -  ((m - 2)/3) x[1]'[
                z] - ((3  m - 
                      1) (m - 2)/(6  m (m - 1))) x[1][
                  z]^2)/Binomial[m + 1, 3]

expr  =  U[2, j]  // FullSimplify

num  =  Numerator[expr]

denom  =  Denominator[expr]

a1 = Coefficient[num,  x[1]'[z]]  // Factor

a2 = Coefficient[num,  x[2][z]]  // Factor

a3 = Coefficient[num, x[1][z]^2]  // Factor

Factor[a1/denom]

Factor[a2/denom]

Factor[a3/denom]