I just read Andrew Robbins' solution to the tetration problem, which I find very convincing, and want to use the opportunity to present and discuss it here.

The solution satisfies the 2 natural conditions

1. and

2. is infinitely differentiable.

However for avoiding difficulties with a later expansion, he instead solves for the tetration logarithm tlog (which he calls super logarithm but I find "tetration logarithm" somewhat more specific), which is the inverse of , i.e. .

The first condition is then translated into

1'. and

while the second condition is equivalent to that also

2'. is infinitely differentiable.

By 1' we merely need to consider the tlog on the interval and can then derive the values for the other intervals , , etc. above and below.

The idea is now that we define a smooth (infinitely differentiable) function t on and ensure that the by 1' resulting function is also smooth at the joining points of the intervals. Obviously it suffices to ensure this for the joining point 0 (and by 1' this transposes to each other joining point).

We simply expand t into a power series at 0 and then try to determine the coefficients such that the resulting function is smooth at 0

is the i-th derivative of t at 0.

The resulting function on is

We have to ensure that for each . What is now the k-th derivative of s at 0?

For we get .

For the constant -1 vanishes and we make the following calculations:

The derivation of is easily determined to be

and so the k-th derivative is , which give us in turn

for .

This is an infinite linear equation system system in the variables .

The way of Andrew Robbins is now to approximate a solution by considering finite linear equation systems consisting of n equations and n variables resulting from letting for .

First one can show that these equation systems have a unique solution for b>1 and numerical evidence then shows that converges and that the resulting are a solution of the infinite equation system.

Further numerical evidence shows, that the infinite sum in the definition of t converges for the so obtained .

However I would guess that the claimed uniqueness for a solution satisfying 1' and 2' is not guarantied. We can use different approximations, for example for a given constant we can consider the equation systems, resulting from letting for . Because interestingly the sum converges to where e is the Euler constant and B_k are the Bell numbers. So by setting for the remaining sum converges and merely introduces an additive constant in the linear equation system. The obtained solutions are different from the solution obtained by c=0.

However I didnt verify yet the convergence properties of these alternative solutions.

The solution satisfies the 2 natural conditions

1. and

2. is infinitely differentiable.

However for avoiding difficulties with a later expansion, he instead solves for the tetration logarithm tlog (which he calls super logarithm but I find "tetration logarithm" somewhat more specific), which is the inverse of , i.e. .

The first condition is then translated into

1'. and

while the second condition is equivalent to that also

2'. is infinitely differentiable.

By 1' we merely need to consider the tlog on the interval and can then derive the values for the other intervals , , etc. above and below.

The idea is now that we define a smooth (infinitely differentiable) function t on and ensure that the by 1' resulting function is also smooth at the joining points of the intervals. Obviously it suffices to ensure this for the joining point 0 (and by 1' this transposes to each other joining point).

We simply expand t into a power series at 0 and then try to determine the coefficients such that the resulting function is smooth at 0

is the i-th derivative of t at 0.

The resulting function on is

We have to ensure that for each . What is now the k-th derivative of s at 0?

For we get .

For the constant -1 vanishes and we make the following calculations:

The derivation of is easily determined to be

and so the k-th derivative is , which give us in turn

for .

This is an infinite linear equation system system in the variables .

The way of Andrew Robbins is now to approximate a solution by considering finite linear equation systems consisting of n equations and n variables resulting from letting for .

First one can show that these equation systems have a unique solution for b>1 and numerical evidence then shows that converges and that the resulting are a solution of the infinite equation system.

Further numerical evidence shows, that the infinite sum in the definition of t converges for the so obtained .

However I would guess that the claimed uniqueness for a solution satisfying 1' and 2' is not guarantied. We can use different approximations, for example for a given constant we can consider the equation systems, resulting from letting for . Because interestingly the sum converges to where e is the Euler constant and B_k are the Bell numbers. So by setting for the remaining sum converges and merely introduces an additive constant in the linear equation system. The obtained solutions are different from the solution obtained by c=0.

However I didnt verify yet the convergence properties of these alternative solutions.