(08/16/2022, 07:35 AM)bo198214 Wrote: 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.