Tetration Forum

The following provides the Schroeder and Abel functions in Mathematica using the support for solving recursive equations and Bell polynomials. The code takes about a hour to run for the first ten derivatives.

Code:
order=10; 

H[0]=0;

H[1]=f'[0]^t ;

Do[H[max]=First[r[t]/.RSolve[{r[0]==0,r[t]==Sum[Derivative[k][f][0]BellY[max,k,Table[H[j]/.t->t-1,{j,max}]],{k,2,max}]+ f'[0] r[t-1]},r[t],t]],{max,2,order}];

Schroeder=f'[0]^t z+Sum[1/k! H[k]z^k,{k,2,order}]

Abel=Limit[Schroeder,{f'[0]->1}]

The Pari/GP code for Bell polynomials is

Code:
  Bell(k,n=-1)=

  { my(x, v, dv, var = i->eval(Str("X",i)));

    v = vector(k, i, if (i==1, 'E, var(i-1)));

    dv = vector(k, i, if (i==1, 'X*var(1)*'E, var(i)));

    x = diffop('E,v,dv,k) / 'E;

    if (n < 0, subst(x,'X,1), polcoef(x,n,'X));

  }

So to Mathematica code at the beginning of this thread can be migrated to Pari/GP.

Daniel

Daniel