\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
I feel like this is overdetermined.
For K = 1,2,3 : let \( \theta_K(z) \) satisfy
1) \( \theta_K(z) = \theta_K(z+1) \)
2) \( \theta_K(n) = 0 \) for every integer \( n \).
3) \( \theta'_K(n) = 0 \) for every integer \( n \).
4) \( z + \theta_K(z) \) is univalent for z near the real line.
5) \( 1 + \theta'_K(z) =/= 0 \) for z near the real line.
Then
\( \alpha_1(\text{tet}_b(z + \theta_3(z) )) = T(z) \)
\( \alpha_2(\text{tet}_b(z + \theta_3(z) )) = T_2(z) \)
Where T(z) and T_2(z) are Taylor series that satisfy T_i(n) = n for all integer n and i.
( Since the LHS is a Taylor series , so is the RHS )
Let \( \alpha_2( \alpha^{[-1]}_1(z) )) = D(z) \)
Then \( D(T(z)) = T_2(z) \)
Now it seems we have more freedom.
However we require - by the above - : \( D(n) = n \)
Fortunately that seems no problems since the the two abel functions agree on integer iterates (alpha_i(b^z) = alpha_i(z)+1 ).
So we end up with :
\( \alpha_1(\text{tet}_b(z + \theta_3(z) )) = T(z) \)
such that \( T(n)=n \)
This seems to be relaxable to
\( \alpha_1(\text{tet2}_b(z)) = T(z) \)
such that \( T(n)=n \)
I suggest z is a positive real to solve the equation.
--------
now if z is a positive real :
\( \alpha_1(\text{tet2}_b(z)) - \alpha_1(\text{tet2}_b(z-1)) = T(z)-T(z-1) \)
And maybe T(z)-T(z-1) is a periodic function ?
--------
***
I notice the resemblance of
\( \alpha_1(\text{tet2}_b(z)) = T(z) \)
towards
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
***
So when are the 5 conditions valid ??
It seems things are simplified.
But I would like to add that the radius of D(z) matters ... or that of D(T(z)).
Equations are only meaningfull within the domain of convergeance.
Maybe that is a criterion to the more general case ( apart from tetration , wheither we can agree upon 2 fixpoints ).
So it seems our attention should go to D and T perhaps rather than tet(z) in order to solve the equation.
Maybe this can be converted into carleman matrices equations.
Still thinking...
regards
tommy1729
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
I feel like this is overdetermined.
For K = 1,2,3 : let \( \theta_K(z) \) satisfy
1) \( \theta_K(z) = \theta_K(z+1) \)
2) \( \theta_K(n) = 0 \) for every integer \( n \).
3) \( \theta'_K(n) = 0 \) for every integer \( n \).
4) \( z + \theta_K(z) \) is univalent for z near the real line.
5) \( 1 + \theta'_K(z) =/= 0 \) for z near the real line.
Then
\( \alpha_1(\text{tet}_b(z + \theta_3(z) )) = T(z) \)
\( \alpha_2(\text{tet}_b(z + \theta_3(z) )) = T_2(z) \)
Where T(z) and T_2(z) are Taylor series that satisfy T_i(n) = n for all integer n and i.
( Since the LHS is a Taylor series , so is the RHS )
Let \( \alpha_2( \alpha^{[-1]}_1(z) )) = D(z) \)
Then \( D(T(z)) = T_2(z) \)
Now it seems we have more freedom.
However we require - by the above - : \( D(n) = n \)
Fortunately that seems no problems since the the two abel functions agree on integer iterates (alpha_i(b^z) = alpha_i(z)+1 ).
So we end up with :
\( \alpha_1(\text{tet}_b(z + \theta_3(z) )) = T(z) \)
such that \( T(n)=n \)
This seems to be relaxable to
\( \alpha_1(\text{tet2}_b(z)) = T(z) \)
such that \( T(n)=n \)
I suggest z is a positive real to solve the equation.
--------
now if z is a positive real :
\( \alpha_1(\text{tet2}_b(z)) - \alpha_1(\text{tet2}_b(z-1)) = T(z)-T(z-1) \)
And maybe T(z)-T(z-1) is a periodic function ?
--------
***
I notice the resemblance of
\( \alpha_1(\text{tet2}_b(z)) = T(z) \)
towards
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
***
So when are the 5 conditions valid ??
It seems things are simplified.
But I would like to add that the radius of D(z) matters ... or that of D(T(z)).
Equations are only meaningfull within the domain of convergeance.
Maybe that is a criterion to the more general case ( apart from tetration , wheither we can agree upon 2 fixpoints ).
So it seems our attention should go to D and T perhaps rather than tet(z) in order to solve the equation.
Maybe this can be converted into carleman matrices equations.
Still thinking...
regards
tommy1729