Just asking...

[bo198214]

So these are 3 equations with 3 variables \( a \), \( b \), \( c \) you can solve them numerically and voila you obtain a polynomial which gives a 2-times differentiable tetration.

The only problem with that approach, I see it now, is that the solutions need not to be real. As it is indeed in this case (the given equation system has only complex solutions). So it seems you have to tell us more about your formula ...