Is this the beta method?

[tommy1729]

http://apminstitute.org/recurrent-abel-f...quation-2/

I just stumbled upon it recently, looks like he describes a superfunction obtained by the beta method?

VERY VERY VERY Similar Bo!

The difference here is that they are choosing very different forms of the limit. This is very similar to how I started approaching the beta method (I wanted a Gaussian kind of representation like the \(\Gamma\)-function). But, this is a rather big but, we are adding in a convergence factor. This author has written a slightly different variation. And additionally, the author is working in the space of continuous functions.

I've found many a continuous version of this theory--I used to communicate to John Gill a fair amount; who described a very similar kind of construction. John Gill did a lot of infinite composition stuff; and is sort of the only real source for a lot of these things. And even then, it is scarce to say the least.

The difference of the beta method; and how I've built up what you see as this \(\mathcal{C}\) notation--is that it is designed for holomorphy. And holomorphy can be found quickly. And it only relies on the convergence of a sum. So the \(\beta\) method, has built into it, the following two theorems:

\[
\beta_\lambda(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\lambda (j-s)} + 1} \bullet z
\]

Is holomorphic everywhere the composite is holomorphic, because:

\[
\sum_{j=1}^\infty |\frac{e^z}{e^{\lambda (j-s)} + 1}|\,\,\text{converges uniformly everywhere the composite is holomorphic}
\]

We can substitute my notation for the \(\mathcal{C}\) notation, yes. The main difference, and largely the difference which separates the \(\beta\) method. Is the description of it in the complex plane. And the fact we can construct these holomorphic functions.

You'll note, that they write the limit of the beta method:

\[
\lim_{N\to\infty} \log^{\circ N} \Omega_{j=1}^N\frac{e^z}{e^{\lambda (j-s-N)} + 1} \bullet z\\
\]

I write it:

\[
\lim_{N\to\infty} \log^{\circ N}\beta_\lambda(s+N)
\]

The big difference here, is that they don't acknowledge there is no holomorphy here in \(s\). It is smooth on \((x_0,\infty)\), but that's it... In the complex plane, this construction converges pointwise, but not uniformly, and is not holomorphic (it's kind of like a devil's staircase).

My form of the limit, is in my opinion superior, because we can take one first, and then the other. And we do so while retaining holomorphy.

You'll actually find a lot of random articles relating to a lot of similar things to the beta method. I am in no way the originator of infinite compositions; or the techniques used to solve these problems. But I have an edge when it comes to complex analysis; and deriving holomorphy of infinite compositions.

TL;DR This is pretty much the beta method, lol. But, as a mathematician, the devil is in the details.

Im still puzzled by the beta method and this post of yours captures all the puzzles ( i think i get all the rest )

First , I think both limit definitions give the same result. Although maybe at different speeds.

Not sure if you accept that or not.

Secondly and more important ;

Are you saying the beta method is not analytic ? near the real line ? or nowhere ?

And proven ? or just conjectured ?

And why ?

sure it has poles.

And one could even argue that those poles can go to infinity if one allows and accepts decreasing bases.

and one has many logs.  but also many branches to pick.

So im still puzzled.

Saying it converges pointwise on the complex plane is even more puzzling to me !


I though a sequence of (bounded) univalent ( locally biholomorphic ) functions that converges , is also analytic in the limit ?

( I write (bounded) because that is somewhat tautologie by the property of convergent )

Isnt that a theorem ??



And how could it converge at the points that were originally poles ??

ln ln ln pole does not converge pointwise ???!!!???



How does one prove the function f(z) is not analytic in lets say

z in  [a + b i , a - b i] 

where a > 1 , b < 1/a ?

***

Forgive me for a somewhat unrelated question that might be better asked elsewhere but 

you talked about infinite compositions as solutions to difference equations.
in particular related to parabolic fixpoints and iterations of z + ...

But i was thinking about a popular equation here 

the binary partition function 

f(n+1) = f(n) + f(n/2)
or
f(n+1) = f(n) + f((n-1)/2)

and similar ones

( such as f ' (x) = f(x/2) , f(x+1) = f(x) + f(x/2) )

so 

what does infinite compositions and " compositional calculus " have to say about it ??

f(s+1) = f(s) + f(s/2)

f(s+1) = id(f(s)) + Q(f(s))

with Q( f(s) ) = f(s/2)

what looks like a superfunction type equation already.

So we have infinite compositions involving

B(z) = z + Q(z)

or similar we investigate the dynamics of iterations of B(z).

We already know alot about the binary partition function but i wonder what this approach can teach us.

see : https://math.eretrandre.org/tetrationfor...hp?tid=911


regards

tommy1729