The purpose of this notebook is to provide an independent check on the correctness of formulas (1.49), (1.53), and (1.59) by showing through direct computations that they specify relative invariants.

The version of Mathematica  used to evaluate this notebook:

$VersionNumber

For the coefficients of (1.1), we have the condition

c[0,0][z_]=1

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

For a transformation (1.2) of (1.1) into (1.3), formula (2.5) on page 23 yields

cStar[i_, j_][z_] = 
      Sum[ Binomial[m - mu, i - mu]*Binomial[m - nu, j - nu]*
          (Derivative[i - mu][rho][z]*
              Derivative[j - nu][rho][z]/rho[z]^2)*
          c[mu, nu][z], {mu, 0, i}, {nu, 0, j}];

where we have written  cStar[i,j][z]  for  \(TraditionalForm\`\(c\^*\)\_\(i, j\)\)(z).   For (1.49), we obtain

Expand[(cStar[1,1][z]-cStar[0,1][z]^2)-(c[1,1][z]-c[0,1][z]^2)]

For (1.53), we  set

Inv[1, 2][m_, z_] = 
      ( c[1, 2][z] - 
          ((m - 1)/m)*c[0, 1][z]*c[1, 1][z] - 
          c[0, 1][z]*c[0, 2][z] + 
          ((m - 1)/m) c[0, 1][z]^3 - 
          ((m - 1)/4) D[c[1, 1][z], z] + 
          ((m - 1)/2) c[0, 1][z]*D[c[0, 1][z], z]);

as well as

InvStar[1, 2][m_, z_] = 
      (cStar[1, 2][z] - 
          ((m - 1)/m)*cStar[0, 1][z]*cStar[1, 1][z] - 
          cStar[0, 1][z]*cStar[0, 2][z] + 
          ((m - 1)/m) cStar[0, 1][z]^3 - 
          ((m - 1)/4) D[cStar[1, 1][z], z] + 
          ((m - 1)/2) cStar[0, 1][z]*D[cStar[0, 1][z], z]);

and obtain Star

Expand[InvStar[1, 2][m, z] - Inv[1, 2][m, z]]

For (1.59), we set

Inv[2, 2][m_, z_] = 
      (c[2, 2][z] - 
          (2 (m - 1)/m) c[0, 1][z]*c[1, 2][z] + 
          (6 (m - 1)/(5 (m + 1) m)) c[0, 2][z]*
            c[1, 1][z] - c[0, 2][z]^2 + 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            c[0, 1][z]^2*c[1, 1][z] + 
          (2 (5  m + 2) (m - 1)/(5 (m + 1) m)) 
            c[0, 1][z]^2*c[0, 2][z] - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            c[0, 1][z]^4 - ((m - 1)/3) D[ c[1, 2][z], z] + 
          ((m - 1)^2/30) D[ c[1, 1][z], {z, 2}] + 
          ((m - 1)^2/(3  m)) c[0, 1][z]*
            D[ c[1, 1][z], z] + 
          ((5  m - 4) (m - 1)^2/(15 (m + 1) m)) 
            D[c[0, 1][z], z]*c[1, 1][z] + 
          ((m - 1)/3) c[0, 1][z]*D[c[0, 2][z], z] + 
          ((m - 1)/3) D[c[0, 1][z], z]*c[0, 2][z] - 
          ((m - 1)^2/15) c[0, 1][z]*D[ c[0, 1][z], {z, 2}] - 
          ((m - 1)^2/15) D[ c[0, 1][z], z]^2 - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m)) 
            c[0, 1][z]^2*D[c[0, 1][z], z]);

as well as

InvStar[2, 2][m_, z_] = 
      (cStar[2, 2][z] - 
          (2 (m - 1)/m) cStar[0, 1][z]*cStar[1, 2][z] + 
          (6 (m - 1)/(5 (m + 1) m)) cStar[0, 2][z]*
            cStar[1, 1][z] - cStar[0, 2][z]^2 + 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            cStar[0, 1][z]^2*cStar[1, 1][z] + 
          (2 (5  m + 2) (m - 1)/(5 (m + 1) m)) 
            cStar[0, 1][z]^2*cStar[0, 2][z] - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            cStar[0, 1][z]^4 - 
          ((m - 1)/3) D[ cStar[1, 2][z], z] + 
          ((m - 1)^2/30) D[ cStar[1, 1][z], {z, 2}] + 
          ((m - 1)^2/(3  m)) cStar[0, 1][z]*
            D[ cStar[1, 1][z], z] + 
          ((5  m - 4) (m - 1)^2/(15 (m + 1) m)) 
            D[cStar[0, 1][z], z]*cStar[1, 1][z] + 
          ((m - 1)/3) cStar[0, 1][z]*D[cStar[0, 2][z], z] + 
          ((m - 1)/3) D[cStar[0, 1][z], z]*cStar[0, 2][z] - 
          ((m - 1)^2/15) cStar[0, 1][z]*
            D[ cStar[0, 1][z], {z, 2}] - 
          ((m - 1)^2/15) D[ cStar[0, 1][z], z]^2 - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m)) 
            cStar[0, 1][z]^2*D[cStar[0, 1][z], z]);

and obtain

Together[InvStar[2, 2][m, z] - Inv[2, 2][m, z]]

Consequently, (1.49), (1.53), and (1.59) specify semi-invariants of the first kind.

To show that (1.49), (1.53), and (1.59) specify semi-invariants of the second kind, we shall use (3.9) on page 30.  For this, we need only the the functions (Subscript[α, 0, j])(ζ ), (Subscript[α, 1, j])(ζ ), and (Subscript[α, 2, j])(ζ ) from (3.1) and (3.2).  Therefore, we begin with

alpha[0][j_, zet_] = 1

alpha[1][j_, zet_] = 
      (Derivative[2][f][zet]/Derivative[1][f][zet])*
        Sum[-k, {k, 1, j}]

alpha[2][j_, zet_] = 
      ( D[ f''[zet]/f'[zet], zet]*
              Sum[-k (k + 1)/2, {k, 1, j}] + 
            (f''[zet]/f'[zet])^2*
              Sum[ (1 + k)^2*k/2, {k, 1, j}]) // Simplify

With

c[0,0][z_]=1

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

formula (3.9) on page 30 gives

cSS[i_, j_][zet_] := 
      Sum[ (f'[zet])^(mu + nu)*alpha[i - mu][m - i, zet]*
            alpha[j - nu][m - j, zet]*c[mu, nu][f[zet]], {mu, 0, i}, {
            nu, 0, j} ] /; (0 <= i <= 2) && (0 <= j <= 2)

for  \(TraditionalForm\`\(\(c\^**\)\_\(i, j\)\)(ζ\ )\).

For (1.49), we obtain

Together[(cSS[1,1][zet]-cSS[0,1][zet]^2)-(f'[zet]^2)*(c[1,1][f[zet]]-c[0,1][f[zet]]^2)]

For (1.53), we  set

\(\(\(Inv[1, 2]\)[m_, z_] = 
      \((\ \(c[1, 2]\)[z]
 - 
          \((\((m - 1)\)/m)\)*\(c[0, 1]\)[z]*\(c[1, 1]\)[z] - 
          \(c[0, 1]\)[z]*\(c[0, 2]\)[z] + 
          \((\((m - 1)\)/m)\) \(c[0, 1]\)[z]^3 - 
          \((\((m - 1)\)/4)\) D[\(c[1, 1]\)[z], z] + 
          \((\((m - 1)\)/2)\) \(c[0, 1]\)[z]*D[\(c[0, 1]\)[z], z])\); \)\)

as well as

InvSS[1, 2][m_, zet_] = 
      (cSS[1, 2][zet] - 
          ((m - 1)/m)*cSS[0, 1][zet]*cSS[1, 1][zet] - 
          cSS[0, 1][zet]*cSS[0, 2][zet] + 
          ((m - 1)/m) cSS[0, 1][zet]^3 - 
          ((m - 1)/4) D[cSS[1, 1][zet], zet] + 
          ((m - 1)/2) cSS[0, 1][zet]*D[cSS[0, 1][zet], zet])
        ;

and obtain

Together[
      InvSS[1, 2][m, zet] - (f'[zet])^3*Inv[1, 2][m, f[zet]]]

For (1.59), we set

Inv[2, 2][m_, z_] = 
      ( c[2, 2][z] - 
          (2 (m - 1)/m) c[0, 1][z]*c[1, 2][z] + 
          (6 (m - 1)/(5 (m + 1) m)) c[0, 2][z]*
            c[1, 1][z] - c[0, 2][z]^2 + 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            c[0, 1][z]^2*c[1, 1][z] + 
          (2 (5  m + 2) (m - 1)/(5 (m + 1) m)) 
            c[0, 1][z]^2*c[0, 2][z] - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            c[0, 1][z]^4 - ((m - 1)/3) D[ c[1, 2][z], z] + 
          ((m - 1)^2/30) D[ c[1, 1][z], {z, 2}] + 
          ((m - 1)^2/(3  m)) c[0, 1][z]*
            D[ c[1, 1][z], z] + 
          ((5  m - 4) (m - 1)^2/(15 (m + 1) m)) 
            D[c[0, 1][z], z]*c[1, 1][z] + 
          ((m - 1)/3) c[0, 1][z]*D[c[0, 2][z], z] + 
          ((m - 1)/3) D[c[0, 1][z], z]*c[0, 2][z] - 
          ((m - 1)^2/15) c[0, 1][z]*D[ c[0, 1][z], {z, 2}] - 
          ((m - 1)^2/15) D[ c[0, 1][z], z]^2 - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m)) 
            c[0, 1][z]^2*D[c[0, 1][z], z]);

as well as

InvSS[2, 2][m_, z_] = 
      (cSS[2, 2][z] - 
          (2 (m - 1)/m) cSS[0, 1][z]*cSS[1, 2][z] + 
          (6 (m - 1)/(5 (m + 1) m)) cSS[0, 2][z]*
            cSS[1, 1][z] - cSS[0, 2][z]^2 + 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            cSS[0, 1][z]^2*cSS[1, 1][z] + 
          (2 (5  m + 2) (m - 1)/(5 (m + 1) m)) 
            cSS[0, 1][z]^2*cSS[0, 2][z] - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m^2)) 
            cSS[0, 1][z]^4 - 
          ((m - 1)/3) D[ cSS[1, 2][z], z] + 
          ((m - 1)^2/30) D[ cSS[1, 1][z], {z, 2}] + 
          ((m - 1)^2/(3  m)) cSS[0, 1][z]*
            D[ cSS[1, 1][z], z] + 
          ((5  m - 4) (m - 1)^2/(15 (m + 1) m)) 
            D[cSS[0, 1][z], z]*cSS[1, 1][z] + 
          ((m - 1)/3) cSS[0, 1][z]*D[cSS[0, 2][z], z] + 
          ((m - 1)/3) D[cSS[0, 1][z], z]*cSS[0, 2][z] - 
          ((m - 1)^2/15) cSS[0, 1][z]*
            D[ cSS[0, 1][z], {z, 2}] - 
          ((m - 1)^2/15) D[ cSS[0, 1][z], z]^2 - 
          ((5  m + 2) (m - 1)^2/(5 (m + 1) m)) 
            cSS[0, 1][z]^2*D[cSS[0, 1][z], z]);

and obtain

Together[
      InvSS[2, 2][m, zet] - (f'[zet])^4*Inv[2, 2][m, f[zet]]]

Consequently, (1.49), (1.53), and (1.59) specify semi-invariants of the second kind of respective weights 2, 3, and 4.  This independent check on the correctness of the typing of formulas (1.49), (1.53), and (1.59) shows that they do specify relative invariants of respective weights 2, 3, and 4.