Generalizing Addition and Multiplication to an Operator Parametrized by a Real Number

Motivation Szekeres Symmetry-Based Questions
Generalization of Addition and Multiplication
to an operator parametrized by a real number
An exploration of fractionally iterated exponentials
Daren Scot Wilson
March 24, 2022

Basic Definition of Multiplication
We all learned as wee young-uns that multiplication is repeated
addition.
A × B = A + A + ... + A
| {z }
B of them

Repeated Multiplication?
So what is repeated multiplication? Powers, exponents and all
that.
A∧
B = A × A × ... × A
| {z }
B of them
Note that I’m using an explicit symbol × for multiplication, not the
common convention of just putting things next to each other, as in AB

But Powers are Ugly!
Addition is symmetric, associative, has an identity element, is
pretty!
x + y = y + x, ∃0 : x + 0 = 0 + x = x, ∃ − x : x + (−x) = 0
Same for multiplication, quite nice!
x × y = y × x, ∃1 : x × 1 = 1 × x = x, ∃x̄ : x × x̄ = 1
But not so for powers, aside from special cases
xy
̸= yx
, x1
= x but no ?y
= y

Really Ugly!
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
but sadly
a(bc)
̸= (ab
)c

What We Want
Can we define an operator ∗ such that:
x ∗ y = y ∗ x, ∃Θ : x ∗ Θ = Θ ∗ x = x, ∃ν(x) : x ∗ ν(x) = 0
and also, observing that multiplication is distributive over
addition,
x × (y + z) = (x × y) + (x × z)
will be distributive over multiplication?
x ∗ (y × z) = (x ∗ y) × (x ∗ z)

Definition of ∗
Yes we can! We’ll skip the details here, but the main idea is to
write out the distributive law, extend it to multiply an arbitrary
number of factors (yes I keep switching what letters to use):
x ∗ (A × A × ... × A)
| {z }
B of them
= (x ∗ A) × (x ∗ A) × ... × (x ∗ A)
| {z }
B of them
x
then set A to Θ. Do some algebra. Eventually we find
x ∗ y = exp(x) ∗ exp(y)/ exp(Θ)
Might as well set Θ = e and so we have
x ∗ y = exp(log x × log y)
or to put it in terms of addition,
x ∗ y = exp exp(log log x + log log y)

Extension
We have +, ×, ∗ and now it seems simple to define the“next”
operator. Let the general operator be denoted ]n with n some
integer.
x ]n y ≡ expn(logn(x) + logn(y))
where
expn(x) = exp(exp(exp(... exp
| {z }
n of them
(x))))
Of course, exp0(x) = x and likewise for log0.

Go Backwards
Heck we could even go backward!
x ⋄ y ≡ x ]−1 y = log(exp(x) + exp(y))
Note that this is distributive over addition:
a + (b ⋄ c) = (a + b) ⋄ (a + c)

Generalizing to Real Parameter
Anytime I see integers from −∞ to ∞ I wonder if we can fill in
the gaps, to generalize integer n to real r.
Can we define an operation halfway between + and ×?
Can we define operations for ANY real number r?
How are inverses defined in all those cases?

Fracionally Iterated Exponentials
We sure can!
As long as we can define fractionally iterated exponential
function for any real order of iteration r.
exp0(x) = x
exp1(x) = ex
expr(x) = exp(expr−1(x))
expr(exps(x)) = exps(expr(x))
logr(x) ≡ exp−r(x)

How to define Fractionally Iterated Exponentials?
So how do we define fractionally iterated exponentials (F.I.E.)in
practice?
One cannot tap the “exp” key on a calculator half a time!

Abel’s Function
One tool to define this beast is Abel’s function, useful for
fractionally iterating any suitable function f():
A(f(x)) = A(x) + 1
For f2 we have
A(f(f(x))) = A(f(x)) + 1 = A(x) + 2
So we generalize:
A(fr(x)) = A(x) + r
fr(x) = A−1
(A(x) + r)
As logarithms map multiplication to addition, Abel’s function maps
iteration to addition.

Non-uniqueness of Abel’s Function
Abel’s function relates fr to fr±1 but does not relate fr with
non-integer spacings, such as fr+.01.
This is similar to generalizing the factorial function to a smooth
real function. Any such function ϕ(x) must obey
ϕ(x) = xϕ(x − 1) but could be replaced by ϕ(x)u(x) where u(x)
is an arbitrary function of period 1. Nothing relates different
values of x except at integer spacing.
By demanding ϕ(x) fit some suitable smoothness or concavity
condition, we find a unique function, the well known Γ function
(shifted over one place).
Γ(x + 1) = x!

Non-uniqueness p. 2
To define our expr uniquely, we must choose one Abel function
out of infinitely many that could do the job.
Given an Abel function A(x) corresponding to iteration of a
function f, we can just as well use
B(x) = p(A(x))
where p is a continuous strictly increasing function that may be
written p(w) = q(w) + w where q is a (nearly) arbitrary
function of period 1.
G. Szekeres, an Australian mathematician, demanded an Abel
function related to expr be smooth in a sense of signs of
derivatives not changing past a certain point.

Szekeres’ g function
Actually, Szekeres worked with a function closely related to the
exponential,
g(x) = ex
− 1
This has a nice property of being “normalized”, g(0) = 0 and
g′(x) = 1.
We want to iterate g,
g0(x) = x, g1(x) = ex−1
gr(gs(x)) = gr+s(x)
and for convenience we’ll define
ḡ(x) ≡ g−1(x) = log(x + 1)
(Actually Szekeres uses e(x) but we’ll soon have too many “e”s running
around, so I use g)

Iteration of g
For very small x, iteration of g is easy - do nothing! g
approximates the identity function.
For larger but still small-ish x, we may approximate gr(x) as a
power series in both r and x. We write:
gr(x) = S0(r)x0
+ S1(r)x + S2(r)x2
+ S3(r)x3
+ ...
By demanding gr(gs(x) = gs(gr(x))) and working out the
algebra, we can define what I’m calling the Szekeres
Polynomials.

Szekeres Polynomials
S0(r) = 0
S1(r) = 1
S2(r) =
1
2
r
S3(r) =
1
4
r2
−
1
12
r
S4(r) =
1
8
r3
−
5
48
r2
+
1
48
r
S5(r) =
1
16
r4
−
13
144
r3
+
1
24
r2
−
1
180
r
...etc...
To check, compute the values of these for r = −1, 0, 1 and verify
you get the usual power series for log(x + 1), x, and exp(x) − 1.

Practical Calculation of Szekeres F.I.E.
Well, this is fine for fractionally iterating the g function, and
only if the value x is small. To iterate exp given any value of x
we make use of the similarity of exp and g for large x.
We map expr to gr by making an intermediate mapping that
identifies expr(x) with gr(x) for “large” x, which we can have
by using expr+m(x) and gr+m(x) for some “large” m ∈ Z In
practice, m might be three or four.
This is explained somewhat better in my paper, and justified by a proper
mathematician in Szekeres[1].

The Szekeres Computational Mountain
To find expr(x) for any x we start with the value of x, apply
exp a few times to climb up to a “large” number, then apply ḡ
a few times to get a “small” value. This takes care of mapping
our desired expr(x) calculation to a practical gr(x) calculation.
Once we’ve done the gr magic using Szekeres polynomials, we
finish by reversing the first mountain climb, climbing up to the
realm of the “large” using g and back down to our original
space using log.

Algorithm: Szekeres F.I.E.
Starting with some value for x, and for −0.5 ≤ r ≤ 0.5,
1. Apply exp until you have a “large” value, from a few
hundred to anything needing scientific notation.
2. Apply ḡ until you have a “small” value, perhaps 0.1 for
quick rough calculations, 0.01 or less for more accuracy.
Call this y for now.
3. Compute gr(y) using Szekeres polynomials.
4. Apply g to the result to reach a “large” value.
5. Apply log enough times so that the
Nexp − Ng + Nḡ − Nlog = 0
If r is not in that range, do more or fewer exp before or log after the
described process.

The Szekeres Computational Mountain Example
Example: exp1/3(4)
Figure: Computing g1/3(4) the Szekeres Way.

Plots of Szekeres F.I.E.
−4 −3 −2 −1 0 1 2 3 4
x
−4
−2
0
2
4
6
8
10
12
exp_u(x)
Szekeres' Fractionally Iterated Exponential
1/2
1/4
1 y=exp(x)
3/4
1/2
1/4
0 y=x
-1/4
-1/2
-3/4
-1 y=log(x)
-5/4

Something Completely Different
A completely different approach... let us draw a column of all
the real numbers, from −∞ to +∞ in a few inches of paper and
ink:
∞

1
0
-1
-∞

Exponentiate
Put exp(x) for each value to the right, making a new column.
The original column is shaded in pink.
∞

∞

1 e
0 1
-1
-∞

0
e
1/
/

See the Full Column
Since the first new column looks like the first, doubled in
length, but we see only the top half, we might extend it like this
∞

∞

1 e
0 1
-1
-∞

0
-1
-e
-∞

e
1/
/
/
e
1/
-

A Bigger Chart
Let’s do exp once more, and extend that column. Also, why not
go the other way? Show logarithms to the left of the original
column.
L
-2 -1 0 1 2 3
∞

∞

∞

∞

∞

∞

-∞

0 1 e ee
eee
N
-∞

0 1 e ee
-1
-∞

0
-1
-e
-∞

e
1/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
e
1/
-
e e
1/
0
16
4
8
12
1
e e
-1
/
ee e
1/
Chart of iterated exp (x)
e
1/
0
e-e
e
e e
1/
ee-e
20
24
28
32 1
e e
-1
/
e

Pattern and Insight
L
-2 -1 0 1 2 3
∞

∞

∞

∞

∞

∞

-∞

0 1 e ee
eee
N
-∞

0 1 e ee
-1
-∞

0
-1
-e
-∞

e
1/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
e
1/
-
e e
1/
0
16
4
8
12
1
e e
-1
/
ee e
1/
e
1/
0
e-e
e
e e
1/
ee-e
20
24
28
32 1
e e
-1
/
e
Notice the nice curves of where zero lies in all the columns, or
all the −∞ or any other values. Something is going on!

Schröder’s Equation
Schröder’s function is similar to Abel’s function but works by
multiplying instead of adding.
L(f(x)) = cL(x)
for some constant c. For arbitrary iteration of f,
L(fr(x)) = cr
L(x)
For our case, f = exp and c = 1/2. The left side of the chart
has λ = L(x) marked.
L
-2 -1 0 1 2 3
∞

∞

∞

∞

∞

∞

-∞

0 1 e ee
eee
N
-∞

0 1 e ee
-1
-∞

0
-1
-e
-∞

e
1/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
e
1/
-
e e
1/
0
16
4
8
12
1
e e
-1
/
ee e
1/
e
1/
0
e-e
e
e e
1/
ee-e
20
24
28
32 1
e e
-1
/
e

Negation in Symmetry-Based F.I.E.
We have placed +∞ and −∞
oppositely about zero, and +1 and −1
oppositely, and as we built the chart,
all values have opposites reflected
about zero. So the zeroth-order
negation operator ν0 is just a simple
reflection.

Reciprocals
The next order inverse, ν1, a.k.a
the reciprocal, is exactly the same
but working on just the upper half
of the chart.

Replicating the Reciprocal
We can’t take logarithms of
negative numbers, but we can use
ν0, then ν1, and finish with
another ν0. In effect, the ν1 can be
considered defined for all real
numbers if we include its ν0
conjugate.

Higher Order Inverses
We can do the same for all orders
of inverses. While defined simply
only over a limited semi-infinite
range of reals, they all can be
conjugated as needed so as to
apply to any real number.

Inverse Hopping
Given any two real numbers A and
B we can relate them with some
series of inverses, conjugated as
necessary.
Such a series is likely to be
infinitely long.
By accepting “close enough” we
can use a finite series of νi.

Algorithm for Symmetry-Based F.I.E.
This inverse hopping leads to a method for computing
fractionally iterated exponentials.
We do exp and log and −x operations in the just the right
order, acting on x while manipulating λ with reflections and
multiplying or dividing by factors of 2, to relate any given real
number x to something simple like −∞ for which we have a
defined value λ = Λ
The magic step that gives us the fractionally iterated exp
doesn’t involve any exotic polynomials or fuss, just a simple
computation of a coefficient, (1/2)r.

Computing λ = L(x)
Given any real number x specified to some accuracy,
1. Start with an empty list S = [ ]
2. Compute x ← log(x) until x goes negative. Keep count.
(Zero if x was already negative.) Append this count to the
list S.
3. Negate: x ← −x
4. Repeat until either x becomes zero, or S becomes “too
long.”
5. Set λ = 1/2
6. Pop the last count off S. Update λ ← λ/2 that number of
times.
7. Reflect λ = 1 − λ.
8. Pop the next value off S and repeat, until S is empty.

Computing x = L̄(λ)
Given λ,
1. Start with an empty list S = [ ]
2. Double λ ← 2λ until it exceeds 1/2. Keep count. (Zero if λ
already large enough.) Append this count to S.
3. Reflect λ = 1 − λ.
4. Repeat until either λ = 1/2 exactly or S becomes “too
long.”
5. Set x = 0.
6. Pop the last value off S. Update x ← exp(x) that many
times.
7. Negate: x ← −x
8. Repeat until list S is empty.

Numerical Example: Finding L
With these two algorithms and Schröder’s function, we compute
the same example as before, exp1/3(4), but using the
Symmetry-based F.I.E.
In finding λ we develop the count list
S = [3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 2, ...]
From this we find, in binary
λ = bin0.00011001001100100100011...
= dec0.098423...
You can write λ in binary by inspection from S

Numerical Example: A New Lambda
We want the 1/3 iteration, so compute a new λ:
λresult =

1
2
1/3
λoriginal
= dec0.0781187..
= bin0.0001001111111111100101100011....]
The list of counts is then
Sresult = [3, 1, 2, 11, 2, 1, 1, 2, 3, 2, 2...]

Numerical Example: Result
Following the algorithm to compute x from this sequence would
require tapping the ‘exp’ key on a calculator eleven times,
leading to an extremely large value.
Upon being negated then exponentiated, this leads to a number
extremely close to zero. While mathematically incorrect, we
could replace this value with exactly zero. Then it’s as if the
list of counts had been
Sapprox = [3, 1, 1]
Completion of the algorithm leads to the final result
exp1/3(4) = 7.37509...

Plot of L(x) - Wide View
−1000 0 1000 2000 3000 4000
x
0.0
0.2
0.4
0.6
0.8
1.0
L(x)
Schröder’s Function, Symmetry-Based

Plot of L(x) - Medium View
−5 0 5 10 15 20
x
0.0
0.2
0.4
0.6
0.8
1.0
L(x)
Schröder’s Function, Main Step

Plot of L(x) - Tighter View
13.0 13.5 14.0 14.5 15.0 15.5 16.0
x
0.050
0.055
0.060
0.065
0.070
0.075
L(x)
Schröder’s Function, Tighter

Plot of L(x) - Detail View
15.18 15.19 15.20 15.21 15.22 15.23
x
0.05738
0.05740
0.05742
0.05744
0.05746
0.05748
0.05750
0.05752
0.05754
L(x)
Schröder’s Function, Detail

L(x) a Fractal?
15.18 15.19 15.20 15.21 15.22 15.23
x
0.05738
0.05740
0.05742
0.05744
0.05746
0.05748
0.05750
0.05752
0.05754
L(x)
It’s like a fractal, but instead of
scaling by some factor in x and y,
we scale in one dimension and
exponentiate in the other.

Extreme Slopes in Schröder’s function
15.18 15.19 15.20 15.21 15.22 15.23
x
0.05738
0.05740
0.05742
0.05744
0.05746
0.05748
0.05750
0.05752
0.05754
L(x)
Note the numerous vertical
segments.

Flat Places, Very Flat
Looking at x = L̄(λ) instead, these
segments are flat, very flat, nearly
constant areas.

Flat Places, Very Flat
Looking at x = L̄(λ) instead, these
segments are flat, very flat, nearly
constant areas.
Pick any very small ϵ, and any very
small range of values of λ, and you’ll
find an infinite number of segments,
with no lower bound in size, with an
absolute value of slope less than that
ϵ.

Flat Places in Binary
λ = 0. 01100101
| {z }
fixed
0000000000
| {z }
run of n
110100011....
| {z }
varying
Define a range of λ values by fixing the first “few” (maybe
zillions) bits in binary.
Then put a long run of n 0’s or 1’s.
After that, anything goes. Varying these bits in all possible
ways defines the range.

Flat Places Origin
λ = 0. 01100101
| {z }
fixed
0000000000
| {z }
run of n
110100011....
| {z }
varying
Computing x from any λ in this range, using the given
algorithm, leads to a nasty series of exp making an extreme
huge number, then after negating and one more exp, a number
astonishingly close to zero. As in the numerical example we did.
This is the origin of the flat places.

Plot of Symmetry-Based Exponentials
−4 −3 −2 −1 0 1 2 3 4
x
−4
−2
0
2
4
6
8
10
12
exp_u(x)
Symmetry-based Fractionally Iterated Exponential
1/2
1/4
1 y=exp(x)
3/4
1/2
1/4
0 y=x
-1/4
-1/2
-3/4
-1 y=log(x)
-5/4

Notes on Plot of Exponentials
−4 −3 −2 −1 0 1 2 3 4
x
−4
−2
0
2
4
6
8
10
12
exp_u(x)
Symmetry-based Fractionally Iterated Exponential
1/2
1/4
1 y=exp(x)
3/4
1/2
1/4
0 y=x
-1/4
-1/2
-3/4
-1 y=log(x)
-5/4
1. Nowhere do the curves cross or
touch.
2. Has infinitely many places with
arbitrarily large slope
3. Has infinitely many arbitrarily
flat places
4. Regular exp1(x) adn log1(x)
are not jagged.

Smoothness Claim
I claim that the functions L(x), L̄(λ), and all expr(x) are
smooth, in the sense of having finite derivatives of all orders at
every point.
(Well, at least where the functions themselves are defined, for
example log(x) exists only for x 0)
Proof? Uh... oh geez look at the time, I got a dentist appointment right
now.. uh.. gotta go!

QUESTIONS
No, this isn’t for you to ask me questions, but for me to give
you a few to think about!

Question: Tiny Order of Iteration
What can we say about expr(x) when r is small?
What are interesting approximations?
expr(x) ≃ f(r, x) where r ∼ 0, x ∼ 0
What can we say about derivatives w.r.t. order of iteration,
d
dr
expr(x) =?

Question: Asymptotic Growth
What approximations can we make for expr(x) for large x?

Question: Asymptotic Growth
What approximations can we make for expr(x) for large x?
How does exp1/2(x) grow as x increases to ∞?

Question: Describing Other Functions’ Growth
When r = 1 we have just ex which we typically don’t try to
approximate, but rather, it is a standard by which we describe
other functions.

Question: Describing Other Functions’ Growth
When r = 1 we have just ex which we typically don’t try to
approximate, but rather, it is a standard by which we describe
other functions.
Are exp1/2(x) or any expr(x) useful for describing the growth of
some (probably weird) functions? Here we have something
that’s greater than polynomial growth, but less than
exponential growth, and different from such beasts as exp(
√
x)
(Naturally it’s ’the Szekeres version we’re talking about here.
Functions similar to the Symmetry-Based one? Hard to
imagine!)

Question: Runless-n Numbers?
Not relating to fractional-order arithmetic operators or
exponentials, but since this came up in our symmetry-based
work...
What if we start with all real numbers, then remove those
whose binary representation has a run of 0’s or 1’s of length n
or longer?
λ = 0. 01100101
| {z }
mixed up
00000000
| {z }
run of 8
110100011....
| {z }
mixed up
What can we say about the set of runless-5 numbers, or
runless-9999?
I smell something like Cantor’s set, or fractals of some sort.

Notes on Runless-n Numbers
All rationals are runless-n for some n.
3/11 = bin0.0101110100010111010001...
The binary for 3/11 contains “000” and “111” but no runs of
four. It’s a member of runless-4.

A Runless-3 Irrational Number
This number is probably transcendental, certainly irrational,
and is runless-3 by design:
Ξ = 0. 10110
| {z }
1pair
10110110
| {z }
2pairs
10110110110
| {z }
3pairs
10110110110110
| {z }
4pairs
...
A single “1” is a marker. The 0’s are like whitespace. Each
marker is followed by k pairs of 1’s before the next marker.
Increment k for each next chunk of bits starting with a marker.

Question: Complex numbers?
Can the values x and y be complex?
Can the order of iteration r be complex?
Hard to see how the symmetry-based F.I.E. could deal with
complex values.
Maybe the Szekeres definition and algorithm, being based on
exp, log and polynomials, has a chance?
Maybe not: the upward ladder of exp was strictly (and quickly)
increasing for real values, but once an imaginary components is
involved, the result of each exp could end up anywhere in the
complex plane, including having large negative real components.
All bets are off on trying to match exp n() and gn() for large n.

Question: Complex Analysis, Riemann Surfaces
Plain simple z is a well-behaved complex function. So is exp(z).
But log(z) is a challenge - it’s multi-valued.
log(z) = log|z| + i arg(z) + 2πin z ∈ C, n ∈ Z
This is well understood in complex analysis, using Riemann
surfaces, making branch cuts when useful.
Assuming the Szekeres F.I.E. can be extended to the full
complex plane, how do we understand exp1/2(z) or log1/2(z)?
How about log.00001(z)?
What fun things can we do with contour integrals for
fractionally iterated logarithms?

References
Szekeres, G. ”Fractional iteration of exponentially growing functions.”
Journal of the Australian Mathematical Society 2, no. 3 (1962): 301-320.
E. Schröder. Ueber iterirte Functionen. Mathematische Annalen 3, 296–322
(1870).
Marek Kuczma. Functional Equations in a Single Variable. Polish Scientific
Publishers, 1968.
Keith Briggs. The work of George Szekeres on functional equations, 2006.
http:
//keithbriggs.info/documents/Szekeres_seminar_QMUL_2006jan10.pdf
Daren Wilson. Fractional Order Arithmetic, J. Undergraduate
Mathematics. Part I (1980) in vol 12 pp. 51-54; Part II (1981) in vol 13
pp.??

A PDF paper covering the same material as this slide deck,
with more detail and full references, along with Python source
code to compute the Szekeres and Symmetry-Based
exponentials, is available on GitHub
https://github.com/darenw/FRITEXP

Generalizing Addition and Multiplication to an Operator Parametrized by a Real Number

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