Quote:@james : nice resume of some of your key ideas.
I even see the connection with fractional derivatives.
Just one question: you say that in many cases the difference equation has no analytic solution ?
like in nowhere analytic ?
Like which ones ?
And in the cases which are not , is it always due to or shown with infinite composition or are there other tools and ideas ?
Does that property relate or carry over to differential equations ??
***
Hey, Tommy. just to clarify some things. Yeah, that was basically a rephrasal of some of the stuff I started to do with Fractional derivatives; but I did more work in analytic number theory where these transforms are much more at home. So this was just a rough explanation of that.
I, also, didn't mean that the difference equation has no analytic solution; I meant that converting the difference equation into an analytic super function proved to have no analytic solution. For example, the \(\beta\) method was spawned from a lot of these observations, and the \(2 \pi i\) periodic tetration base \(b = e\) is nowhere analytic. So the \(\beta\) function itself is analytic, but it can't spawn an analytic superfunction in this instance (it becomes \(C^\infty\) on the real line and a tad more chaotic elsewhere (but an equivalent kind of \(C^\infty\))).
For the most part it was derived using infinite compositions, but I used a lot of tools I had gathered over the years from complex analysis.
When you mention differential equations; we kind of go off topic from this discussion. But much of this "difference equation" talk can be turned into "differential equation" talk.
For example. the general solution to a difference equation:
\[
\Delta f = q(s,f)\\
\]
Has the form:
\[
f(s) = \Omega_{j=1}^\infty z + q(s-j,z)\bullet z\\
\]
Where \(z\) acts as an initial point parameter. Very similar to Picard lindelof. This solution is unique if we ask that \(\lim_{s \to -\infty} f(s) = z\)--and then it only converges for certain \(z\). We can extend this to differential equations by solving arbitrary difference equations:
\[
\begin{align}
f(s+h) - f(s) &= h q(s,f(s))\\
f(s) &= \Omega_{j=1}^\infty z + q(s-jh,z)h\bullet z\\
\end{align}
\]
Limiting \(h \to 0\) produces the general form of a first order differential equation. This gets very very fucking complicated though, which led me to posit the Compositional integral. Which is designed after the Riemann-Stieljtes integral. I won't bother going into detail. But if you're interested I have a short over view on Arxiv; and then I have what I consider my thesis on Compositional integration--which describes all the methods and the ways these objects can converge. I'd suggest the overview though; as the full thesis is far far more indepth and dealt almost exclusively with compositional integration as though it was cauchy's contour integration.
This was intended to be a notice to some people at U of T, and I was planning on publishing it more professionally, but then covid happened and I can't be bothered anymore, lol.
https://arxiv.org/abs/2001.04248
For example, if you write:
\[
f(s,z) = \lim_{h\to 0} \Omega_{j=1}^\infty z + e^{(s-jh)z}h\bullet z\\
\]
Then this function satisfies:
\[
\begin{align}
f'(s,z) &= e^{sf(s,z)}\\
\lim_{s\to-\infty} f(s,z) &= z\\
\end{align}
\]
In the compositional integral notation, this would be written:
\[
f(s,z) = \int_{-\infty}^s e^{tz}\,dt\bullet z\\
\]
And this object converges everywhere:
\[
\int_{-\infty}^s ||e^{tz}||_{z \in K} \, |dt| < \infty
\]
Where \(K\) is compact. Incidentally it means the solution is holomorphic for \(s \in \mathbb{C}\) and \(\Re(z) > 0\).
The overview was basically a motivation for the notation, to describe how it works, where it comes from. It has nothing to do with difference equations. I only touched briefly on the "difference equations become differential equations" in an adjacent paper, as it became pretty self explanatory once you have strong normality theorems. The thesis I wrote, dealt much more with this stuff, where I looked at developing fourier transforms. Where you have results like:
\[
\begin{align}
\int_{-\infty}^\infty z f(t)e^{-2 \pi i t\xi}\,dt \bullet z &= z e^{\int_{-\infty}^\infty f(t)e^{-2\pi i t\xi}\,dt}\\
\int_{-\infty}^\infty z^2 f(t)e^{-2 \pi i t\xi}\,dt \bullet z &= \frac{1}{\frac{1}{z} + \int_{-\infty}^\infty f(t)e^{-2\pi i t\xi}\,dt}\\
\end{align}
\]
And these are invertible Fourier transforms. This extends for general functions \(p(s,z)\) and not just \(p(s,z) = g(z)f(s)\)--but doing so becomes a problem much like Tate's thesis on fourier transforms in abstract algebra and the likes. It's basically useless for tetration, but has it's value elsewhere.
