Welcome Jayd,
yes you observed correctly: the 3 properties
1. \( {}^1 b = b \)
2. \( {}^{x+1} b = b^{{}^{x}b} \)
3. and infinite differentiability (even analyticity)
do not suffice to uniquely determine an extension of tetration.
See also my post about Andrew's solution where I describe modified solutions also following from the approach of defining piecewise an infinite differentiable solution.
They suffice neither for the Gamma function nor for exponentiation nor for multiplication. The criterion that makes the Gamma function unique is logarithmic convexity and the criterion that makes exponentiation and multiplication unique is the translation equation (see the FAQ for a description of both).
So - as you said - for uniqueness there shall be found a suitable criterion, (however I beleave there exists nothing suitable XD).
For the fractional iteration of functions there are uniqueness criterions at hand if the function has a fixed point. Unfortunately this is not the case for exp, indeed there is no uniqueness criterion for continuous iteration of exp too. (For the relationship between iteration of exp and tetration see also the FAQ.)
Can you explain your formula
\( T(x,\ y,\ n) = \left{
\begin{eqnarray} \alpha_0\ +\ y\ +\ x^{T(x,\ y-1,\ n-1)} & , & n\ >\ 0 \\ \alpha_0\ +\ y & , & n\ =\ 0 \end{eqnarray} \right.
\\ \
\\ \
\\ \
\\
{\Large ^y x}\ =\ \lim_{m,n\to\infty}{ln^{\small (m)}T(x,m+y,m+n)} \)
in a bit more detail?
PS: For not using TeX before your post was amazing
yes you observed correctly: the 3 properties
1. \( {}^1 b = b \)
2. \( {}^{x+1} b = b^{{}^{x}b} \)
3. and infinite differentiability (even analyticity)
do not suffice to uniquely determine an extension of tetration.
See also my post about Andrew's solution where I describe modified solutions also following from the approach of defining piecewise an infinite differentiable solution.
They suffice neither for the Gamma function nor for exponentiation nor for multiplication. The criterion that makes the Gamma function unique is logarithmic convexity and the criterion that makes exponentiation and multiplication unique is the translation equation (see the FAQ for a description of both).
So - as you said - for uniqueness there shall be found a suitable criterion, (however I beleave there exists nothing suitable XD).
For the fractional iteration of functions there are uniqueness criterions at hand if the function has a fixed point. Unfortunately this is not the case for exp, indeed there is no uniqueness criterion for continuous iteration of exp too. (For the relationship between iteration of exp and tetration see also the FAQ.)
Can you explain your formula
\( T(x,\ y,\ n) = \left{
\begin{eqnarray} \alpha_0\ +\ y\ +\ x^{T(x,\ y-1,\ n-1)} & , & n\ >\ 0 \\ \alpha_0\ +\ y & , & n\ =\ 0 \end{eqnarray} \right.
\\ \
\\ \
\\ \
\\
{\Large ^y x}\ =\ \lim_{m,n\to\infty}{ln^{\small (m)}T(x,m+y,m+n)} \)
in a bit more detail?
PS: For not using TeX before your post was amazing