11/30/2022, 04:45 AM
(11/30/2022, 02:06 AM)Daniel Wrote:(11/30/2022, 01:13 AM)JmsNxn Wrote:(11/28/2022, 12:37 PM)Daniel Wrote: I need to understand umbral calculus better as I believe there is a neat representation of iterated functions there.
Skip umbral calculus and go straight to Ramanujan. I've studied both, and umbral calculus is just a naive, early, little rigor, version of Ramanujan's work. Ramanujan, singlehandedly, showed how right umbral calculus was, but additionally did it with new tools. Upon which, more rigorous mathematicians soldified all of Ramanujan's observations. A lot of umbral calculus is "coincidences we can't explain"--Ramanujan gave the explanation.
JmsNxn, I'm really up for hearing more about Ramanujan's approach. I have volume 2 and 5 of his notebooks. As far as umbral calculus, my understanding is that Rota put it on solid ground using operator theory and Sheffer sequences. Down with binomial hegemony, liberate all the calculus'.
Ramanujan's approach to everything he did, was analysing sums as integral transforms. His work on modular forms, for example, involves a complicated transformation of a discrete sequence, which when summed, can be integrated in such a manner, that you have 50% of the theory of modular forms. When it comes to umbral calculus; Ramanujan, and people near this idea gave credence to a lot of the classical knowledge. I mean, umbral calculus was always right, but it was never proven (especially the basic ideas).
For example. Let's take a holomorphic function \(f(z) : \{\Re(z) > 0\} \to \{\Re(z) > 0\}\). Now, let's take a common expansion in umbral calculus, which does work. Let's write:
\[
\begin{align}
\Delta f(z) &= f(z+1) - f(z)\\
\Delta^n f(z) &= \Delta \Delta^{n-1}f(z)\\
\end{align}
\]
Let's write now \((z)_n\) as the pochammer symbol, wherein:
\[
\Delta (z)_n = n(z)_{n-1}\\
\]
Where from here, we write one of the cornerstones of Umbral calculus:
\[
f(z) = \sum_{n=0}^\infty \left(\Delta^n f(0)\right)\frac{(z)_n}{n!}\\
\]
This is what's known as a Newton series. It shouldn't be knew to many people. But actually proving this thing exists and converges is a difficult thing.
Now, this is a defining identity of much of umbral calculus (and if you have this, many of the traditional umbral results follow similarly). The trouble is, this "newton series" is very hard to sum, and it's unclear when it is summable. For this, we turn to Mellin/Ramanujan -- Hell, even Riemann used similar ideas. (I think one of the greatest problem with this series, especially throughout history, is that it does converge; but it converges really god damn slow.)
Ramanujan's identification between the Mellin transform and difference equations, continues this in a much better way. We start by saying \(f(z) = O(e^{\rho|\Re(z)| + \tau|\Im(z)|})\) and we restrict \(|\tau| < \pi/2\). Which means we have exponential growth, but the imaginary growth is less that \(\pi/2\). Then, by nature of Ramanujan, there exists:
\[
\vartheta(x) = \sum_{n=0}^\infty f(n)\frac{(-x)^n}{n!}\\
\]
Where this object is holomorphic. And additionally:
\[
\int_0^\infty \vartheta(x)x^{z-1}\,dx = \Gamma(z) f(-z)\\
\]
Where \(\Gamma\) is Euler's Gamma function. From this sole manipulation, we can construct Newton's series, and not only that; justify it rigorously. Let's write:
\[
\mathcal{F} \vartheta = \frac{1}{\Gamma(z)}\int_0^\infty e^{-x} \vartheta(x)x^{z-1}\,dx\\
\]
This can be writ:
\[
\frac{d^{-z}}{dx^{-z}}\Big{|}_{x=0} e^{-x}\vartheta(x)\\
\]
If I write:
\[
\mathcal{F}\frac{d}{dx}\vartheta(x) = \Delta\mathcal{F}\\
\]
This should be apparent. If I write:
\[
\mathcal{F}x\vartheta(x) = s \mathcal{F}\left\{\vartheta(x)\right\}(s+1)\\
\]
This should equally be apparent.
We get the beautiful identity:
\[
\frac{1}{(n-1)!} \int_0^\infty e^{-x}\vartheta(x)x^{n-1}\,dx = \Delta^{n} f(0)\\
\]
So now, let's do the straightforward transformation:
\[
\begin{align}
f(-z) &= \frac{1}{\Gamma(z)} \int_0^\infty \vartheta(x)x^{z-1}\,dx\\
&= \frac{1}{\Gamma(z)} \int_0^\infty e^xe^{-x}\vartheta(x)x^{z-1}\,dx\\
&= \sum_{n=0}^\infty \frac{1}{n!} \frac{1}{\Gamma(z)}\int_0^\infty e^{-x}\vartheta(x)x^{n+z-1}\,dx\\
&= \sum_{n=0}^\infty \Delta^n f(0) \frac{(-z)_n}{n!}\\
\end{align}
\]
Now, this stuff has been around since the 18th century. The integral transforms are old. But Ramanujan really set it in stone. Quite literally, Daniel; it's just a \(e^{x} e^{-x} = 1\), while expanding these things underneath an integral. There is still some work to do here, but once you have an expansion for a newton series; you can do everything in umbral calculus.
All I'm trying to say.
Happy to keep talking, I have not read the references you suggested. But nothing less than love
