Mathematica for Sections 6.1 and 6.2 (evaluated)

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Section 6.1

a[m_,1][z_]:=(1/Binomial[m+1,3])(w[2][z]-((m-1)/2)w[1]'[z]-((m-1)/(2m))w[1][z]^2)

d[m_,1][z_]:=(1/(m(m-1)))w[1][z]

K[m_,1,i_,j_][z_]:=0/;i≤-1

K[m_,1,0,j_][z_]:=1

K[m_,1,i_,j_][z_]:=(Sum[(D[K[m,1,i-1,k][z],z]-(m-1)*d[m,1][z]*K[m,1,i-1,k][z]+(m+2-i-k)(2-i-k)a[m,1][z]*K[m,1,i-2,k][z]),{k,j+1,m}])/;i≥1

w[0][z_]=1;X[k_][z_]:=w[k][z]

L[m_,1,i_][z_]:=Sum[K[m,1,i-j,j][z]*X[j][z],{j,0,i}]

M[m_,1,e1_,i_][z_]:=FunctionExpand[Binomial[m-i,e1-i]]*Product[(e1-r),{r,1,e1-i}]*L[m,1,i][z]

A[e1_,i_]:=-1/(e1+i-1)/;i≥1

B[e1_,i_]:=(e1-i)/(e1+i-2)/;i≥1

inv[m_,1,e1_,0][z_]:=0

inv[m_,1,e1_,1][z_]:=0

inv[m_,1,e1_,i_][z_]:=(M[m,1,e1,i][z]+A[e1,i-1]*D[inv[m,1,e1,i-1][z],z]+B[e1,i-1]*a[m,1][z]*inv[m,1,e1,i-2][z])/;i≥2

basicInv[m_,1,e1_][z_]:=inv[m,1,e1,e1][z]

Expand[basicInv[3,1,3][z]]

list[e1_]:=Table[D[w[j][z],{z,k}],{j,1,e1},{k,0,e1-j}]//Flatten

Collect[basicInv[m,1,3][z],list[3],Factor]

Collect[basicInv[4,1,3][z],list[3],Factor]

Collect[basicInv[4,1,4][z],list[4],Factor]

Section 6.2

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

d[m_,1][z_]:=(1/(m(m-1)))w[1][z]

U[m_,1,i_,j_][z_]:=0/;i≤-1

U[m_,1,0,j_][z_]:=1

U[m_,1,i_,j_][z_]:=(Sum[D[U[m,1,i-1,k][z],z]+2(i-1+k-m)d[m,1][z]*U[m,1,i-1,k][z]+(m-i-k+2)(2-i-k)b[m,1][z]*U[m,1,i-2,k][z],{k,j+1,m}])/;i≥1

w[0][z_]=1;X[k_][z_]:=w[k][z]

V[m_,1,i_][z_]:=Sum[U[m,1,i-j,j][z]*X[j][z],{j,0,i}]

W[m_,1,e1_,i_][z_]:=FunctionExpand[Binomial[m-i,e1-i]]*Product[(e1-r),{r,1,e1-i}]*V[m,1,i][z]

A[e1_,i_]:=-1/(e1+i-1)/;i≥1

B[e1_,i_]:=(e1-i)/(e1+i-2)/;i≥1

jnv[m_,1,e1_,0][z_]:=0;jnv[m_,1,e1_,1][z_]:=0

jnv[m_,1,e1_,i_][z_]:=(W[m,1,e1,i][z]+A[e1,i-1]*(D[jnv[m,1,e1,i-1][z],z]+2(i-1)*d[m,1][z]*jnv[m,1,e1,i-1][z])+B[e1,i-1]*b[m,1][z]*jnv[m,1,e1,i-2][z])/;i≥2

basicJ[m_,1,e1_][z_]:=jnv[m,1,e1,e1][z]

list[e1_]:=Table[D[w[j][z],{z,k}],{j,1,e1},{k,0,e1-j}]//Flatten

Collect[basicJ[m,1,3][z],list[3],Factor]

Collect[basicJ[m,1,4][z],list[4],Factor]

diff[e1_]:=Together[basicInv[m,1,e1][z]-basicJ[m,1,e1][z]]

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Do[Print["diff[",e1,"] = ",diff[e1]],{e1,3,9}]

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