continuation of fix A to fix B ?

[tommy1729]

Consider a general entire function f(z) with a repelling fixpoint A and an attracting fixpoint B such that :

1) there is at least one path from A to B by iterations of f(z).

2) the other fixpoints are relatively far away from A and B and the path(s) , or they do not exist.

3) B is actually 0.

4) 0 < norm(f ' (B)) <  norm(1/c)  < 1 

**

Then we can solve the schröder equation

sch( f(z) ) = (1/c) sch(z)

by

sch(z) = sum_k  c^k * f^[k](z)

where sum_k means summing over all integer k.

Notice this is not , or not necc , the koenings function ;

I mean not the koenings function developped at point A or point B.

Afterall we know the koenigs does not agree on fixpoints and here we have 2.

On the other hand we only have a part of the neighbourhood of the fixpoints and not the fixpoints themselves agreeing.

But a koenings function should work all around the ( complete ) neighbourhood of its expansion fixpoint.
SO agreeing on 2 fixpoints * and their neighbourhoods * is problematic.

proof is trivial ;

sch(z) /c  = sum_k  c^(k-1) * f^[k](z) =  sum_k  c^k * f^[k+1](z) = sch(f(z))



Now the main question :

Can we extend by summability methods and analytic continuation , in a logical and consistant way , the function sch(z) beyond the path(s) from A to B ? 

***

Note that the associated superfunction has the same period as c^z , although there are branches etc.

***

Now I noticed that if we have good methods for summability and analytic continuation for double exp functions than

if f(z) is a polynomial ,

We can probably solve this problem for those functions.

***

the idea is that we end up with a superfunction T(z) such that T( - oo + z ) = A AND T( + oo + z) = B = 0.

and a potential 3rd fixpoint lies towards T(+ oo i ) and/or T ( - oo i )

.

Since the koenigs satisfies the semi-group homom near its fixpoint expansion points , this function cannot do the same near its fixpoints ,

so in THAT region it does not satisfy the addition semi-group homom ... but maybe elsewhere it does ...
... unless it is a rule that summability and ana continuation forbid the local creation of the addition semi-group homom ??

Another question is if this ( locally AT LEAST ?? ) agrees with the carleman matrix method expanded somewhere ??

In particular an expansion point on such a path from A to B ? OR maybe rather NOT on that path ? 

I considered connections to the gaussian method and the fibonacci numbers , and somewhat green's theorem and convolution ideas but not with a clear result or understanding ...

( i concluded that the infinitesimal ideas i recently had reduce to nothing usefull , consistant or new ideas , only to arrive at old ideas. So that is why im silent about them now )

I could say or speculate more but this will do for now.

What are your ideas ?

Regards 

tommy1729

[tommy1729]

I want to point out that we can redirect the fixpoints.

So we are not limited to fixpoints being 0.

So this methods works in quite general cases.

More specific it can be applied for tetration bases between 1 and exp(1/e).

So it is an slog method for those bases.

And I started to think and conjecture about their derivatives.

Completely monotonic and such ideas...

regards

tommy1729

[JmsNxn]

Hey, Tommy

I haven't fully digested this. But let me see if I get you straight.

Let's write:

\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]

Now, to make your example concrete, let's put some domains in question. Let's let \(\Im(z) > 0\), where we assume some kind of convergence on \(z \in \mathbb{R}\), and in an analytic manner. Now, let's play your trick, but slightly different. As is always fun, it's better to stick to what we know first. So let's let \(f(z) = \log_{\sqrt{2}}(z+4)-4\) (I promise you, this is going some where). Now, famously, this value tends to \(0\), and for most of the upper half plane does so. By which:

\[
A = \sum_{k=0}^\infty c^k \left(\log_{\sqrt{2}}^{\circ k}(z+4)-4\right)
\]

The values \(\log_{\sqrt{2}}^{\circ k}(z+4)-4 = O(1/2\log(2))^k\)

This value converges for all \(0 < |c| < 2\log(2) > 1\). Now, let's take the negative:

\[
B = \sum_{k=1}^\infty c^{-k} \left(\exp^{\circ k}_{\sqrt{2}}(z+4) - 4\right)\\
\]

Which looks like:

\[
\exp^{\circ k}_{\sqrt{2}}(z+4) - 4 = -2 + O(\log(2)^k)\\
\]

Therefore, this object converges for \(|c| >1\).  Therefore in the tiny annulus, of \(1 < |c| < 2\log(2)\) we've constructed:

\[
F(c,z)
\]

Such that:

\[
F(c,\log_{\sqrt{2}}(z+4)-4) = F(c,z)/c\\
\]

And \(z\) is fairly well behaved, it's at least convergent on the interval \((2,4)\), but I suspect much much larger--I can write the exact domains for you if you'd like. It's the \(\beta\) function ...

---

NOW! I went into this confidently because I knew it'd converge. Because it is the statement of Levenstein's equation in my paper on the beta method. We can actually solve many Weird and Strange Schroder equations. The only unique one, is the one with the multiplier about a fixed point.

For example, I've solved the Schroder equation here, so long as \(1 < |c| < 2\log 2\).

Normally I would write this as follows:

\[
\varphi_\lambda(e^{\lambda}z) = f(\varphi_\lambda(z))\\
\]

Whereby, \(\Re(\lambda) > \log\log(4)\). This is the same thing you've written and discovered. Though we need to do a fair amount of variable changes. I do have uniqueness conditions on the various types of Schroder functions, and what can and can't be a Schroder function. I believe I called it Levenstein's theorem (Or the generalized Levenstein theorem, in my paper). Which is essentially a statement that this thing can only be holomorphic in the annulus \(1 < |c| < 2 \log 2\).


WHAT I WILL SAY IS THIS IS A CRAZY REPRESENTATION! I've tried similar things, but I guess it never clicked. So congratulations Tommy, you've rediscovered a lot of what Levenstein was writing about about the beta method, and the reconstruction of Schroder functions!!!! And you've got a cool expansion to boot!  

I had to use infinite compositions to drag out this result; you've done it with sums. I feel embarassed


Also, I opted to calling this Levenstein's equation; as he ran so much code, and really brute forced everything. And determined, for example with:

\[
f(z) = \log_\sqrt{2}(z)\\
\]

That we can only make a Schroder equation with multiplier \(0 < |c| < \frac{1}{2\log(2)}\). You may have heard me talk about how we can "shrink the period of the Schroder iteration, but we can't grow the period". This is the exact same thing.

But this is an awesome formula for the "Inverse Levenstein Equation" (Levenstein and I worked solely on the "inverse Schroder"). It's the exact same thing though.

AWESOME WORK, TOMMY!

[JmsNxn]

To further exemplify the relation to what I called Levenstein's equation, I'll start from scratch here.

Let's write:

\[
g_\lambda(e^{\lambda}w) = \frac{wf(g_\lambda(w))}{w+e^{\lambda}}
\]

Now, let's take the iteration:

\[
f^{\circ -n} \left(g_\lambda(e^{\lambda n}w)\right) = \varphi_\lambda(w)\\
\]

It became apparent very early on to sheldon, that there was a restriction on \(\lambda\) which allowed this object to converge, versus diverge. And additionally, there was no modification to the steps which could salvage it. I proved that Sheldon's guess was 100% correct--though I framed it as a result of "where \(\beta\) does converge, and where it can't".

Where by the function:

\[
\varphi_\lambda(e^{\lambda} w) = f(\varphi_\lambda(w))\\
\]

And this function will be holomorphic in a half plane in \(\lambda\), and holomorphic in \(w\) near the fixed point--but never in a neighborhood of the fixed point (there will be singularities near the fixed point).

GREAT WORK TOMMY! Your sum can be analytically continued to mine and Sheldon's approach, so this is fucking gold! Thanks a lot, Tommy!


EDIT:

I also called this the Mock Schroder Equation, because it was Schroder's equation with different multipliers.

Okay, this is even weirder! I think this is like the Complementary Levenstein, I'm too tired right now. Need to look at this tomorrow, lmao. But Great job again, Tommy!


I think you've just analytically continued \(\varphi_\lambda\)!

[JmsNxn]

I am officially submitting this function before tommy.


\[
F_\epsilon(c,z) = \sum_{k=-\infty}^\infty c^k e^{-\epsilon k^2}f^{\circ k}(z)
\]

Which is actually the integral:

\[
\int_{-\infty}^\infty c^ke^{-\epsilon k^2}f^{\circ k}(z)\,d\mu(k)
\]

Where \(\mu\) is a measure of moles. Where it has dirac deltas at the integers.

We can write \(F_\epsilon \to F\) as \(\epsilon \to 0\)--We can do this better than I said before. But there's another trick I'm hiding; which looks like:

\[
G(s,z) = \sum_{k=-\infty}^\infty e^{-\epsilon(s-k)^2}f^{\circ k}(z)\\
\]

Which satisfies \(G(s+1,z) = G(s,f(z))\).

We can do this a lot of ways. I'm happy to give you 80% of the credit for some of the things I'm about to do, Tommy. Raes method, would be how I'd say it. I think it aligns very well with everything you've been doing lately.

Not trying to steal your thunder, just trying to get in on it

---

TO BEGIN:

\[
G(s,z) = \int_{-\infty}^\infty e^{-\pi(s-k)^2}f^{\circ k}(z)\,d\mu(k)
\]

Then, this is actually a fourier transform:

\[
e^{-\pi(s-k)^2} = e^{-\pi s^2} e^{2\pi s k}e^{-\pi k^2}
\]

If we move the variable \(s\) to \(-is\); then we get a Fourier looking expansion.

\[
G(-is,z) = e^{\pi s^2}\int_{-\infty}^\infty e^{-2\pi i s k}e^{-\pi k^2}f^{\circ k}(z)\,d\mu(k)
\]

Now, a little rearrangement:

\[
e^{-\pi s^2} G(-is,z) = \int_{-\infty}^\infty e^{-2 \pi i s k} e^{-\pi k^2}f^{\circ k}(z)\,d\mu(k)
\]

Right hand side is a fucking fourier transform!!!!!!
 Fecking ya!

The inverse then, is that:

\[
e^{-\pi k^2}f^{\circ k}(z) = \int_{-\infty}^\infty e^{2 \pi i s k} e^{-\pi s^2} G(-is,z)\,d\widehat{\mu}(s)
\]

I'm a little lazy at the moment to find \(\widehat{\mu}\); but a rough memory is that in the above instance \(\widehat{\mu} = \mu\) (EDIT: THIS IS WRONG! \(\widehat{\mu}\) is the indicator function on \([0,1]\)). When we choose the integers as our moles/indicators/dirac shit, then it's idempotent. But if not, and I'm misremembering. For any measure \(\mu\), there is a Fourier transform dual \(\widehat{\mu}\)--and that's what this notation means.

---

Fuck

So Essentially. Tommy, you have sent me down a path where I believe I can finally show the following result.

Let:

\[
H(s,z) = \sum_{k=-\infty}^{\infty} e^{-\pi(is+k)^2} f^{\circ k}(z)
\]

Then:

\[
e^{-k^2}f^{\circ k}(z) = \sum_{j=-\infty}^\infty e^{-\pi j^2} e^{-2\pi i k j} H(j,z)
\]

EDIT: THIS IS WRONG. IT SHOULD BE:

\[
e^{-k^2}f^{\circ k}(z) = \int_0^1e^{-\pi j^2} e^{2\pi i k j} H(j,z)\,dj
\]

Again, I might have screwed up some things; but fuck this is it:

Fuck

Raes Method

[JmsNxn]

Okay, I'm going to approach this a tad more level headed, because I truly believe Tommy is on to something. And I have a sneaking suspicion this analytically continues the Levenstein Schroder function (which is rampant in Sheldon's code), to a larger domain. Or we have some kind of discontinuity when connecting them. If this is an analytic continuation, I think it's absolutely pivotal.

I'm going to change some language here, and start from scratch. So to begin, we are going to fix \(|c| > 1\). We are going to let \(f(z) = \sqrt{2}^{z+2}-2\)--and then we're going to write the first expansion, Tommy has demonstrated.

\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]

Now, for convenience, assume that \(z \in (0,2)\), or exists in some small neighborhood of this line, without intersecting \((-\infty,0)\) or \((2,\infty)\), which are respective julia sets of \(f^{-1}\) and \(f\). We know that:

\[
f^{\circ k} = O(\log 2 ^k)
\]

For \(z\) in the attracting basin of \(f(0) = 0\). (which the domain I mentioned belonged to). And we know that:

\[
f^{-1}(z) = \log_{\sqrt{2}}(z+2) - 2\\
\]

So that \(f^{-1}\) has an attracting fixed point at \(2\), which looks like \(f^{-1}(z) = 2 + \frac{z-2}{2\log(2)} + O ((z-2)^2)\); and therefore it is attracting at a geometric rate \(\frac{1}{(2\log(2))^k}\) to the fixed point \(2\). Therefore the series:

\[
\sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]

Is holomorphic for \(1 < |c| < \frac{1}{\log(2)}\) and at least a neighbhorhood of each point \(z \in (0,2)\). This constructs, what I will call the Raes coordinate, or Raes expansion.

Now I want to take a look at something a bit different and show a striking resemblence.

---

Sheldon was the first to do this change of variables, and utilized it greatly when constructing his program to evaluate the \(\beta\) method. It's an aesthetic change more than anything, but still greatly valuable. Let's write:

\[
g_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\bullet z\\
\]

This is the unique solution to the equation:

\[
g_c(cw) = \frac{wf(g_c(w))}{w+1}\\
\]

Now, if we take:

\[
f^{\circ -k} (g_c(c^k w)) = \Psi_k(c,w)\\
\]

Then this object only converges for \(|c| > \frac{1}{\log(2)}\). And the limit function satisfies:

\[
\Psi(c,cw) = f(\Psi(c,w))\\
\]

This absolutely DOES NOT have an obvious inverse function in the neighborhood of a fixed point. Iconically, it is solving the inverse schroder equation, but with an arbitrary multiplier (though the domains are inverted from Tommy's version).

---

So, in my humble opinion. We have a GREAT new result. Which I will walk through.

If \(f\) is holomorphic at zero, with a geometric fixed point here; and in the immediate basin \(A_0\), the iterates \(f^{\circ -k} \to L\), another fixed point (\(L\) can be infinite, but that'll require more bounding stuff)--then we can construct a Raes function \(R(z)\) such that:

\[
R(f(z)) = cR(z)\\
\]

If and only if \(1 < |c| < \frac{1}{|f'(0)|}\)


Now, to contrast this with the Levenstein function.... If \(f\) is holomorphic at zero, with a geometric fixed point here--then we can construct a Levenstein function \(L(z)\), such that:

\[
f(L(z)) = L(cz)\\
\]

If and only if \(|c| > \frac{1}{|f'(0)|}\).

The question then is, if we write \(R(c,z)\) and \(L(c,z)\)... IS \(R(c,z)\) an analytic continuation of \(L^{-1}(c,z)\). I'm willing to be this is true. Where a bunch of singularities arrive on the circle \(|c| = \frac{1}{|f'(0)|}\).

---

To get a good idea what's happening here, let's switch from Levenstein, to Beta. So let's write \(c = e^{\lambda}\), where now Raes' function looks like:

\[
R_\lambda(z) = \sum_{k=-\infty}^\infty e^{\lambda k} f^{\circ k}(z)\\
\]

Where now, \(R_\lambda(f(z)) = e^\lambda R_\lambda(z)\), and:

\[
L_\lambda(e^\lambda z) = f(L_\lambda(z))\\
\]

Where then, if we write:

\[
F_\lambda(s) = L_\lambda(e^{\lambda s})\\
\]

We have:

\[
F_\lambda(s+1) = f(F_\lambda(s))\\
\]

Where this is holomorphic for \(\Re\lambda> - \log\log 2\).

If we write:

\[
F_\lambda(s) = R^{-1}_\lambda(e^{\lambda s})\\
\]

Then this is holomorphic for \(0 < \Re \lambda < -\log\log 2\).

This then allows us to say...

There exists a holomorphic tetration \(F_\lambda(s)\) for \(\Re \lambda \neq -\log\log 2\) and \(\Re\lambda > 0\). Which means, not only can we shrink the period with the beta method... we can grow it!



HOLY FUCK NICE JOB TOMMY!!!!!!!!

These things should definitely be continuations of each other, but I'm not seeing at the moment how. Things can get tricky, because we need to identify where the singularities of the Raes function are. It's no surprise that \(F_\lambda(s)\) should have singularities at \(2\pi i/\lambda + j\) for \(\Re(\lambda) > -\log\log 2\). But this would require a change of variables in some way shape or form to map where the singularities are for the Raes function. This will have something to do with theta mappings, but for the life of myself I can't see it obviously.

Quite clearly we need to show there's a theta function \(\theta_\lambda\), which is holomorphic for \(\Re\lambda > 0\) with singularities somewhere on \(\Re \lambda = -\log \log 2\). And show that, using the normal Schroder function:

\[
F_\lambda(z) = \Psi^{-1}\left( e^{\log\log 2( z + \theta_\lambda(z))}\right)\\
\]

Where \(\theta_\lambda(z+1) = \theta_\lambda(z)\) and \(\theta_\lambda(z+2\pi i/\lambda) = \theta_\lambda(z) - 2\pi i/\lambda\).

Jesus, this looks like it's gonna be a headache ....


I think Elliptic functions is going to save us here! But I need to dust off some old books. Essentially, I think I can prove Tommy's theta function for his Raes function is the same as the beta function UPTO an elliptic function. Finding the elliptic function is gonna be a bitch and a half. But I think I see how we might get it. Fuck!!!

This is one for the books Tommy!!!! You've successfully constructed a function \(F_\lambda(z)\) such that \(\sqrt{2}^{F_\lambda(z)} = F_\lambda(z+1)\) and \( 0 < \Re \lambda < -\log \log 2\). It is real valued for \(0 < \lambda < -\log \log 2\) and it has period \(2 \pi i / \lambda\). And additionally, it is highly non trivial! Have a glass of champagne on me! !!!!!

---

OH FUCK! I SEE IT NOW!!!

So I forgot a negative sign! We can prove absolutely that Raes coordinate is an analytic continuation of the Levenstein coordinate. This is really nuanced though, and you gotta let me work through some shit! YES! We can have \(F_\lambda(s)\) for \(\Re(\lambda) > 0\) with singularities only on the line \(\Re\lambda = -\log\log 2\). I forgot that \(R_\lambda(f(z)) = e^{-\lambda}R_\lambda(z)\) where we use \(1/c = e^{-\lambda}\) rather than \(c = e^{\lambda}\)--this changes everything. Much of the discussion is still pretty good, but I forgot a whole layer of complexity

[tommy1729]

Dear James

Thank you for your kind words.

I am not sure what paper or result you refer to with your mention of mock theta and co work with sheldon.

I guess you mean this paper :

https://arxiv.org/pdf/2108.13519.pdf

But I am not sure.

I did not find sheldon or mock theta mentioned.

Or maybe it was not on arxiv ?

It would probably help if we knew what paper you references towards.

Maybe some links here.

Happy new year.

Regards

tommy1729

[JmsNxn]

Hey, Tommy

I'll point you to this report:

https://arxiv.org/pdf/2208.05328.pdf

And specifically Section 2 On Mock Abel And Mock Schroder coordinates.

I only analytically prove the Mock Abel coordinate becoming an Abel coordinate here, but they can be transformed into a mock Schroder equation by a change of variables. You'll probably have to read the whole report to get a good grasp on what I mean by the Weak Julia set, and the Weak Fatou set.

But, just a quick refresher.

A mock Abel equation looks like:

\[
F(s+1) = f(F(s)) h(s)\\
\]

(Here, I always use a logistic function \(h(s) = \frac{1}{1+e^{-\lambda s}}\) for some \(\Re \lambda > 0\))

Where \(h(s) \to 1\) as \(\Re(s) \to \infty\). And a Mock Schroder equation looks like:

\[
P(c w) = f(P(w)) h(\log_c(w))\\
\]

(I've set \(c = e^\lambda\))....

Then, in this paper, I show that on the Weak Fatou set; we have that:

\[
G(s) = \lim_{n\to\infty} f^{-n} F(s+n)\\
\]

Converges--and similarly:

\[
H(w) = \lim_{n\to\infty} f^{-n} P(c^n w)\\
\]

Converges, on the weak fatou set.

What happens that Sheldon was the first to notice, is that if \(|c| \ge |f'(0)|\), this object has a nontrivial domain of convergence, and the process I had written fails for \( 0 \le |c| \le |f'(0)|\). And it appeared that no such function should exist--least ways, my method of construction fails...

---

The weak Fatou set is characterized as follows: If \(s_0\) is in the weak Fatou set, then there exists an open neighborhood \(\mathcal{N}\) about \(s_0\) such that

\[
\limsup_{n\to \infty} \sup_{s\in\mathcal{N}}\frac{1}{|F(s+n)|} < \infty\\
\]

And if \(s_0\) is not in the weak Fatou set, it's in the weak Julia set. The weak Fatou set is open, and the weak Julia set is closed, and they partition the complex plane. This definition is equivalent to:

\[
\limsup_{n\to\infty} \sup_{s\in\mathcal{N}} \frac{1}{|f^{\circ n}(F(s))|} < \infty\\
\]

And if we make the change of variables \(s = \log_c(w)\), you have to account for this in the definition, but it works the same manner.

For example, letting \(\beta(s+1) = \sqrt{2}^{\beta(s)}/(1+e^{-s})\) produces this fractal for the weak Julia set:

   

Which appears at every point \(s = (2k+1)\pi i + j\) for \(k,j \in \mathbb{Z}\). The white fractal is the weak Julia set, and the black area is the weak Fatou set.

This means that:

\[
\lim_{n\to\infty} \log_{\sqrt 2} \beta(s+n) = G(s)\\
\]

converges uniformly in the black area, and nowhere in the white area (which is sort of the central thesis of this report). And this creates a tetration that satisfies \(G(s+1) = \sqrt{2}^{G(s)}\) and \(G(s + 2\pi i) = G(s)\).  Towards the end of this report I describe how this induces a theta mapping \(\theta(s+1) = \theta(s)\); where the regular tetration \(\text{tet}_{\sqrt{2}}(s)\) can be turned into \(G\) by the equation \(G(s) = \text{tet}_{\sqrt{2}}(s + \theta(s))\). Aditionally \(\theta\) is very close to an elliptic function. It's 1 periodic, and linear in a second period. Which is given in this instance as \(\theta(s+2 \pi i) = \theta(s) - 2\pi i\).

---

Mind you, this paper only deals with exponential functions \(e^{\mu z}\) for \( \mu \neq 0\), but it extends for any function. There's nothing special about the exponential function being used. So if we were to switch this to arbitrary functions, nothing really changes; especially near attracting/repelling fixed points. I just wrote the work for tetration

[tommy1729]

Thank you for your useful insightful reply James.

I wanted to add that we can use the same trick for 0 or 1 fixpoint as well.

The 0 fixpoint case is a bit complicated and relates to quite old posts i made.
Usually mixing sums and limits and sometimes summability methods.
That case might not even be analytic.

However I wanted to share the beauty of the 1 fixpoint case.

WLOG lets set the fixpoint at 0 to make things easy.

sketch of the idea :

I will use the example f(x) := 5x + x^3 for which we want to find the super and abel function.
( and thereby we get close to taking a half-iterate , but there is a theta function issue )

Finding the superfunction S(x) such that S(x+1) = f(S(x)) follows by using the classical koenigs function around the fixpoint 0.


So we are left to find

F(x) such that F(f(x)) = c F(x)

( from which by taking log_c(x) we find the abel function ... notice however this abel might not be the inverse of the koenigs function !! Hence not necc giving the half-iterate by combining this abel with the koenigs function. aka the theta function issue mentioned above )

notice f ' (0) = 5.

Also notice n iterates of f(x) grow about x^(3^n)

So we get for appropriate c :

F(x) = sum_k c^k asinh( f^[k](x) )

where the sum runs over all integer k.

notice asinh is close to id(x) near 0.

and asin(x^(3^n)) is close to 3^n.

so if for instance c is around 1/4,

our sum behaves a bit like sum_n  (4/5)^n + (3/4)^n 

where the sum runs over positive integers n.

hence it converges !

and we get F(x) such that F(f(x)) = c F(x).

So we see this idea is part of a large family.


I guess you like that.



regards

tommy1729

[JmsNxn]

Yes, Tommy!

That's very much my point. This finds the complimentary solution. I think we can work a tad differently though; so let me take your example but flip it slightly on its head. Let:

\[
f(z) = 5z + z^3\\
\]

Now define the function:

\[
\gamma_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\,\bullet z\\
\]

For \((|c| > 1\). This function is holomorphic in \(c\) and \(w\), so long as \(w \neq -c^j\) for \(j \ge 1\). And further, is a meromorphic function \(\gamma_c(w) : \mathbb{C}^2 \to \widehat{C}\). This function satisfies:

\[
\gamma_c(cw) = \frac{w \gamma_c(w)}{w+1}\\
\]

Now, we can take the iteration:

\[
\Psi_c(w) = \lim_{k\to\infty} f^{-k} \gamma(c^k w)\\
\]

This iteration only converges for \(|c| > 5\).

My point is that, if we look at, instead, your summation condition, (like the one you wrote), we should be able to solve if for \(1 < |c| < 5\)...

The uniqueness of these equations is difficult to describe--and it requires understanding the uniqueness of these almost elliptic functions \(\theta(z+1) = \theta(z)\) and \(\theta(z+2 \pi i/\log( c)) = \theta(z) - 2\pi i/\log(c )\). These functions are unique UPTO an elliptic function and the location of the singularities and their singular parts.

Suppose:

\[
\theta_1 \neq \theta_2\\
\]

Suppose every where \(\theta_1\) has a singularity \(z_0\), so does \(\theta_2\). Suppose additionally, the singular parts of this singularity are equivalent (this is difficult to describe properly, so I'll use a weaker condition at the moment):

\[
\lim_{z \to z_0}|\theta_1(z) - \theta_2(z)| < \infty\\
\]

Then, \(\theta_1 = \theta_2 + C\) for a constant \(C\), because:

\[
\theta_1(z) - \theta_2(z) = \wp(z)
\]

And:

\[
\wp(z+1) = \wp(z)\,\,\,\,\wp(z+2\pi i/\log( c)) = \wp(z)\\
\]

The function \(\wp(z)\) has no singularities. Therefore it is bounded on \(\mathbb{C}\). By Liouville's theorem, therefore it is constant.

So essentially, any solution to these types of equations, only differ by the singular behaviour of their theta function... Which equates to the statement that they are equivalent upto an elliptic function!