Bifurcation of tetration below E^-E

[andydude]

Wow, You found it!

Aside from the proof you gave, I also did some tests just to be sure. Here is the Mathematica code I used:

Code:

In[1]:= f[k_, y_] := Exp[ProductLog[k, Log[y^y]]/y]

In[2]:= FullSimplify@Table[Limit[D[f[0, y], {y, k}]/k!, y -> 1/E, Direction -> 1], {k, 0, 4}]

Out[2]= {E^(-E), 0, -E^(3 - E)/6, E^(4 - E)/18, (E^(5 - E)*(-82 + 15*E))/1080}

In[3]:= FullSimplify@Table[Limit[D[f[-1, y], {y, k}]/k!, y -> 1/E, Direction -> -1], {k, 0, 4}]

Out[3]= {E^(-E), 0, -E^(3 - E)/6, E^(4 - E)/18, (E^(5 - E)*(-82 + 15*E))/1080}

and as you can see, these coefficients are the same as those I derived on page 2 of this thread, so this is definitely the function! Good Job GFR!

Andrew Robbins