Complex Tetration and Ackermann Function

[Daniel]

See http://tetration.org/tetration_net/tetra...torics.htm and http://tetration.org/tetration_net/tetra...omplex.htm for my solutions for complex tetration. The combinatorics information can be used for solutions for complex Ackermann function.

[bo198214]

See http://tetration.org/tetration_net/tetra...torics.htm and http://tetration.org/tetration_net/tetra...omplex.htm for my solutions for complex tetration. The combinatorics information can be used for solutions for complex Ackermann function.

If I understand it right you use the uniqueness of continuous iteration of formal powerseries with a fixed point. To achieve this you need a fixed point of \( b^x \) and call it \( a_0 \).
Then you continuously iterate \( \exp_b(x)=b^x \) expanded at this fixed point and define then tetration as \( {}^xb=\exp_b^{\circ x}(1) \).
This is an interesting approach and makes me ponder about the possible fixed points, solutions of \( b^x=x \). Because \( \sqrt[x]{x} \) has its maximum at \( x=e \) we conclude that exactly for \( b=e^{1/e} \) (interestingly this base occurs also in Jayd's approach) there is exactly one real fixed point \( a_0=e \) of \( b^x \).

Yes this is striking. Do you have also some Mathematica/Maple/etc code prepared? Is convergence guarantied? It would be quite interesting to compare it with the solutions of Andrew and Jayd.
However because there is no base transform formula for tetration, it maybe that \( e^{1/e} \) is the only base with a certain uniqueness.

Note also that Kneser used a similar approach to define the continuous iterates of \( e^x \), he however used a complex fixed point (as we have seen for \( b=e \) there is no real fixed point). And the result was not real valued, so made some manipulations to make it real valued, see

[1] H. Kneser, Reelle analytische Lösungen der Gleichung \( \varphi(\varphi((x)) = e^x \) und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56–67 (German).

(Note: in the coming days I will make a post about continuous iteration of powerseries with fixed points.)

[andydude]

Interesting you mention base-(\( e^{1/e} \)) tetration, because I also beleive that this base is the only base for which convergence can be proven, although convergence of series for other bases might in fact converge.

Also, there are essentially 3 types of series expansions of iterated exponentials (one for each argument). There is Galidakis' expansions (around the hyper-base), Geisler's expanions (around the main argument), and my expansions (around the hyper-exponent, but inverse). So naturally these are very difficult to compare, if that is indeed what you want to do.

Most general iteration expansions involve series expansions around the main argument (x), where the coefficients are functions of t (iteration/time), but the easiest way to solve functional equations usually involves a series in t whose coefficients are functions of x. So if AR_b(z) is my expansion and DG_bt(x) is Geislers extension, and IG_t(b) is Galidakis' extension, they could be compared as:

• series-transpose to b of DG_bt(1) = IG_t(b)
  
• series-transpose to t of DG_bt(1) = series-inverse of (AR_b(z)=t) = AR^-1_b(t)
  

Andrew Robbins

[Daniel]

I have Mathematica resources at http://tetration.org/Resources/Files/Mathematica/ . SchroederSummations.nb implements what I call Schroeder Summations which is the current basis of my work with the complex Ackermann function. Convergence is predicated on using the proper simplification for the geometrical progressions which results in the different types of fixed points (hyperbolic, parabolic, ...). The parabolic case b=e^(1/e) simplifies to a polynomial expansion giving uniqueness where all other cases simplify to expansions based on the powers of the log of the fixed point.

[bo198214]

While graphing the continuous iterations of the just discussed \( F(x)=e^{x/e} \), they looked quite divergent. Of course the numerics can be quite errournous for our rapidly increasing functions. But now indeed I found a proof of the divergence of \( F^{\circ t}(x) \) for nearly any t.

Proof: Assume that \( F^{\circ t}(x) \) has a convergence radius \( >0 \) for some t developed at the fixed point e.
Then consider the linear transformation \( \tau(x)=(x+1)e \) and its inverse \( \tau^{-1}(x)=\frac{x}{e}-1 \) and define

\( G:=\tau^{-1}\circ F\circ \tau \),

i.e.

\( G(x)=\frac{\left(e^{\frac{1}{e}}\right)^{(x+1)e}}{e}-1=\frac{e^{x+1}}{e}-1=e^x-1 \)

Further it is clear that \( G^{\circ t}=\tau^{-1}\circ F^{\circ t}\circ \tau \). If \( F^{\circ t}(x) \) has a convergence radius at \( e=\tau(0) \) then \( G^{\circ t}(x) \) has a convergence radius at \( 0 \). However it is well known (see [1]) that the (unique) continuous iteration \( G^{\circ t}(x) \) of \( e^x-1 \) has a convergence radius at 0 merely for integer \( t \), see [1].

[1] I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z 69, 1958, 121-163.

[andydude]

Interesting, this is the same technique I use in http://math.eretrandre.org/tetrationforu...d=31#pid31.
I guess it was pretty obvious after all.

Andrew Robbins

[bo198214]

A function \( G=\tau^{-1}\circ F\circ \tau \) is called a conjugate of the Function \( F \), where \( \tau \) can be an arbitrary other function. With respect to iteration, where we know that \( G^{\circ t}=\tau^{-1}\circ F^{\circ t}\circ \tau \), one always tries to reduce the iteration to the iteration of a simpler function.

For example it was found by Scheinberg [1], that each formal powerseries \( f=\sum_{i=1}^\infty f_i x^i \) is conjugate to \( ax+bx^n+cx^{2n-1} \) for appropriate complex \( a,b,c \) and natural \( n \).

[1] Scheinberg, Power series in one variable, J. Math. Anal. Appl. 31 (1970), 321-333.

[UVIR]

See http://tetration.org/tetration_net/tetra...torics.htm and http://tetration.org/tetration_net/tetra...omplex.htm for my solutions for complex tetration. The combinatorics information can be used for solutions for complex Ackermann function.

If I understand it right you use the uniqueness of continuous iteration of formal powerseries with a fixed point. To achieve this you need a fixed point of \( b^x \) and call it \( a_0 \).
Then you continuously iterate \( \exp_b(x)=b^x \) expanded at this fixed point and define then tetration as \( {}^xb=\exp_b^{\circ x}(1) \).
This is an interesting approach and makes me ponder about the possible fixed points, solutions of \( b^x=x \). Because \( \sqrt[x]{x} \) has its maximum at \( x=e \) we conclude that exactly for \( b=e^{1/e} \) (interestingly this base occurs also in Jayd's approach) there is exactly one real fixed point \( a_0=e \) of \( b^x \).

Yes this is striking. Do you have also some Mathematica/Maple/etc code prepared? Is convergence guarantied? It would be quite interesting to compare it with the solutions of Andrew and Jayd.
However because there is no base transform formula for tetration, it maybe that \( e^{1/e} \) is the only base with a certain uniqueness.

Note also that Kneser used a similar approach to define the continuous iterates of \( e^x \), he however used a complex fixed point (as we have seen for \( b=e \) there is no real fixed point). And the result was not real valued, so made some manipulations to make it real valued, see

[1] H. Kneser, Reelle analytische Lösungen der Gleichung \( \varphi(\varphi((x)) = e^x \) und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56–67 (German).

(Note: in the coming days I will make a post about continuous iteration of powerseries with fixed points.)

This method seems very similar to what this guy has in mind in this page.

I tried using his method to extract a Taylor expansion, but got nowehere. Can anybody make sense of what he means? Daniel?