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### The Analytic Continuation of the Ackermann Function

### Exponential Generating Functions

What lies beyond exponentiation?

Extending the arithmetic operations beyond addition, multiplication, and exponentiation to the complex numbers.

Overview

Very high level overview because of the amount of material in multiple branches of mathematics.

- Complex Systems – A New Kind of Science
- Arithmetic
- Dynamics of the Complex Plane
- Combinatorics

New Kind of Science

- Chaos beyond exponentiation.
- Vertical catalog of complex systems.
- Based on iterated functions.
- Arithmetic and physics are two major roles played by iterated functions.
- Iterated functions as a candidate for a fundamental dynamical system in both mathematics and physics.

Arithmetic

- Arithmetic is part of the Foundations of Mathematics.
- Ackermann function is a recursive function that isn’t primitively recursive.
- Different definitions of the Ackermann function.
- Transfinite mathematics

Systems of Notation for Arithmetic Operators

Operator

Spiral

Ackermann

Knuth

Conway

Addition

a+b

ack(a,b,0)

Multiplication

a*b

ack(a,b,1)

Exponentiation

ab

ack(a,b,2)

a ↑ b

a→b→1

Tetration

ba

ack(a,b,3)

a ↑↑ b

a→b→2

Pentation

ba

ack(a,b,4)

a ↑↑↑ b

a→b→3

Hexation

ack(a,b,5)

a↑↑↑↑b

a→b→4

...

...

...

Circulation

ack(a,b,∞)

a ↑∞ b

a→b→∞

Definition of Ackermann Function

Let f(x) ≡ a → x → k and f(1) = a → 1 → k = a;

then

f2(1) = f(a) = a → a → k

= a → 2 → (k+1)

f3(1) = f(a → a → k) = a → (a → a → k) → k

= a → 3 → (k+1)

Negative Integers

a → 1 → 2 = a

a → 0 → 2 = 1 This differs from the historical Ackermann function where a → 0 → 2 = a

The next two are multivalued so the values on the principle branch are shown.

a → -1 → 2 = 0

a → -2 → 2 = -∞

Period three behavior

-1 → 0 → 3 = 1

-1 → 1 → 3 = -1

-1 → 2 → 3 = -1 → -1 → 2 = 0

-1 → 3 → 3 = -1 → (-1 → 2 → 3) → 2 = -1 → 0 → 2 = 1

The first indication that for negative integer the Ackermann function can be very stable.

forn>2

-1→k→n = - 1 → (k+3) → n.

-1 → 0 → n = 1

-1 → 1 → n = -1

-1 → 2 → n = 0

-1 → 3 → n = 1

Transfinite Ackermann Expansions

a → ∞ → k useful for creating a series of “interesting” transfinite number.

Transfinite nature of circulation:

2 → 2 → ∞ = 4

2 → 3 → ∞ = 2 → (2 → 2 → ∞) → ∞

= 2 → 4 → ∞

= 2 → (2 → (2 → 2 → ∞) → ∞) → ∞

= 2 → (2 → 4 → ∞) → ∞

= ∞

3 → 2 → ∞ = 3 → 3 → ∞

= 3 → (3 → 3 → ∞) → ∞

= 3 → (3 → (3→3→∞) → ∞) → ∞

= ∞

Tetration

- First objective is understanding tetration.
- What if tetration and beyond is vital for mathematics or physics?
- With so many levels of self organization in the world, tetration and beyond likely exists.

Dynamical Systems

- Iterated function as a dynamical system.
- Analytic continuation can be reduced to a problem in dynamics.
- Taylor series of iterated function. Most mathematicians believe this is not possible, but my research is consistent with other similar research from the 1990’s.
- Iterated exponents for single valued and iterated logarithms for multi-valued solutions.

Fa di Bruno formula

- Hyperbolic case
- Maps are Flows

- Derivatives of composite functions.
- Fa di Bruno difference equation.

Classification of Fixed Points

- Topological Conjugancy and Functional Equations – Multiple Cases for Solution
- Fixed Points in the Complex Plane
- Superattracting
- Hyperbolic (repellors and attractors)
- Irrationally Neutral
- Rationally Neutral
- Parabolic Rationally Neutral

Combinatorics

- OEIS – On Line Encyclopedia of Integer Sequences
- Umbral calculus and category theory.
- Bell polynomials as derivatives of composite functions.

Dm f(g(x))

- Schroeder summations.
- Hierarchies of height n and the combinatorics of tetration.

Hierarchies of 2 or Bell Numbers

Hierarchies of 3

Hierarchies of 4

xe - Tetration as phylogenetic trees of width x

Validations

- Deeply consistent with dynamics.
- f a (f b(z)) - f a+b(z) = 0 verified for a number of solutions.
- Software validates for first eight derivatives and first eight iterates. Mathematica software reviewed by Ed Pegg Jr.
- A number of combinatorial structures from OEIS computed correctly including fractional iterates.

NKS Summary

- Wolfram’s main criticism is inability of continuous mathematics to deal with iterated functions.
- CAs are mathematics not physics, many non-physical solutions.
- “Physics CA” needs OKS for validation.
- CAs appear incompatible with Lorenz transforms and Bell’s Theorem.

Summary

- Subject is in protomathematics stage, but becoming acceptable areas of research; numerous postings on sci.math.research lately.
- Arithmetic → Dynamics → Combinatorics → Arithmetic
- If maps are flows, then the Ackermann function is transparently extended.
- Suggests time could behave as if it is continuous regardless of whether the underlying physics is discrete or continuous.
- Continuous iteration connects the “old” and the “new” kinds of science. Partial differential iterated equations
- Tetration displays “sum of all paths” behavior, so logical starting place to begin looking for tetration in physics is QFT and FPI. Tetration and many other iterated smooth functions appear compatible with the Lorenz transforms and Bell’s Theorem.

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