Chapter 10, Section 10.3.2, Figure 10.10

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]]

Set up a list consisting of the values of j, α and the function of α for values of j between 2 and 50.

data =
Table[
{m, alpha=N[((m-1)/m)^(2/3)];alpha,
(alpha/(m-1))*
(2*m*LaplaceCoefficient[1/2,m,alpha,0]+
 alpha*LaplaceCoefficient[1/2,m,alpha,1])},
{m,2,50}];

Display the results as a table.

TableForm[data]

Transpose the data ...

tdata = Transpose[data];

... and select the values of j and the function of α for the plot.

sdata = Transpose[{tdata[[1]],tdata[[3]]}];

Generate the plot.  (See Fig.10.10.)

ListPlot[sdata, PlotRange->{{0,50},{1.49,1.61}}, Frame->True]