Chapter 8, Section 8.5, Table 8.5

Use the definition of LaplaceCoefficient from the DisturbingFunction package.

LaplaceCoefficient[s_,jj_Integer,a_,n_Integer]:=
  Module[{outside,j=Abs[jj],cterm,cbterm,csum,i,k},
    outside=2*Product[s+k,{k,0,j-1}]/j!*a^(j-n);
    cterm=1;
    cbterm=cterm*Product[j-k,{k,0,n-1}];
    For[i=1; csum=cbterm,(cbterm>$MachineEpsilon)||(cbterm==0),i++,
      cterm=cterm*(s+i-1)*(s+j+i-1)/i/(j+i)*a^2;
      cbterm=cterm*Product[j+2*i-k,{k,0,n-1}];
      csum=csum+cbterm];
    Expand[outside*csum]]

Define a function, AlphaValues, to return α, α(Subscript[f, 1])(α) and α(Subscript[f, d])(α) for each resonance specified by p+q and p.

AlphaValues[{ppq_, p_}] :=
Module[{j=ppq, α = N[(p/(ppq))^(2/3)]},
f1 = 
 (1/8)*(2*α*LaplaceCoefficient[1/2, 0, α, 1] +
        α^2*LaplaceCoefficient[1/2, 0, α, 2]);
If[ppq-p == 1, 
fd =
 (1/2)*(-2*j*LaplaceCoefficient[1/2, j, α, 0] +
        -α*LaplaceCoefficient[1/2, j, α, 1])];
If[ppq-p == 2,
fd =
 (1/8)*((-5*j + 4*j^2)*LaplaceCoefficient[1/2, j, α, 0] +
        (-2*α + 4*j*α)*LaplaceCoefficient[1/2, j, α, 1] +
         α^2*LaplaceCoefficient[1/2, j, α, 2])];
{ppq, p, α, α*f1, α*fd}
]

Select the resonances for the table.

resonances = {{2,1}, {3,2}, {4,3}, {5,4}, {6,5},
              {3,1}, {5,3}, {7,5}, {9,7}, {11,9}};

Generate the values for the table.

table = Map[AlphaValues[#]&, resonances];

Display the results.

TableForm[table]