Check on Theorem 27.7 of page 313 (Memoir-Jan-05), for n = 1

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

p  =  10;

TrueQ[ $VersionNumber >= 3.0]

$Version Number

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

w[0][z_]=1;

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

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

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[i1_]:=Sum[ K[i1 - r1, r1]*w[r1][z],{r1,0,i1}]

H[l1_, h1_] := 
      Binomial[m - h1, l1 - h1]*Product[(l1 - r), {r, 1, l1 - h1}]

M[l1_,i_]:=Evaluate[(H[l1, h1]*L[h1] /. h1 -> i)]

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}]

V[i1_]:=Sum[U[i1 - r1, r1]*w[r1][z],{r1,0,i1}]

W[l1_,i_]:=Evaluate[(H[l1, h1]*V[h1] /. h1 -> i)]

A[l1_,i_]:=-1/(l1 + i - 1)

B[l1_,i_]:=(l1 - i)/(l1 + i - 2)

difference[l1_]:=Module[{t, u},t[0] = 0;t[1] = 0;Do[t[i] = 
                      M[l1, i] + A[l1, i - 1]*D[ t[i - 1], z] + 
                        B[l1, i - 1]*a[2][z]*t[i - 2],{i, 2, l1}];Inv[l1] = t[l1];u[0] = 0;u[1] = 0;Do[u[i] = 
                      W[l1, i] + 
                        A[l1, i - 1]*
                          (D[u[i - 1], z] + 
                              (i - 1)*b[1][z]*u[i - 1]) + 
                        B[l1, i - 1]*b[2][z]*u[i - 2],{i, 2, l1}];J[l1] = u[l1];Print["The number of terms in Inv[", l1, "] is ", 
                  Length[Expand[Inv[l1]]]];Print["The number of terms in J[", l1, "] is ", 
                  Length[Expand[J[l1]]]];
Print["The memory currently used (in bytes) is ", 
                  MemoryInUse[]];Print["Inv[", l1, "] - J[", l1, "] = ", 
                  Expand[Inv[l1] - J[l1]]];Print["The maximum memory used (in bytes) is ", 
                  MaxMemoryUsed[]];Print[Date[]];Print["     "];Clear[t, u, Inv, J]]

Do[difference[l1],{l1,3,p}]