(12/10/2022, 10:07 PM)MphLee Wrote: Posts like this by James are really gold, I'd like to collect them and make some pdf, so that they will be saved by possible problems on the forum. There is alot to study here.
Right now I just can make a quick superficial comment: this makes me think of this conversation we had one year ago about the analogy sums:integral=omega notation:composition integral.
This makes me wonder if we can add the missing columns of the analogy
\(\displaystyle \sum / \int \,\sim\, \Omega / \int ...\bullet z\)
\(\Delta/ D \,\sim\, ?? / ??\)
\(n! / n! \,\sim\, ?? / ??\)
\(2^x / e^x \,\sim\, ?? / ??\)
\((x)_n / x^n \,\sim\, ??/??\)
\(Newton/ Taylor \, \sim \, ??/??\)
\(\Delta^n/ {\frac{d^z}{d^zx}} \,\sim\, ?? / ??\) fractional calculus?
Oh God, Mphlee! Don't even get my motor running on this. It just hurts my head thinking about it. There's definitely something like this. But you're trying to build the roof of a house before you have the walls, lmao
Don't distract me with what could be! lmao! I only ever got a rough fourier transform for the compositional integral, I cannot do this with Ramanujan's theorem! Would definitely be super cool! And there's definitely something like this somewhere in there.(12/10/2022, 10:11 PM)Daniel Wrote: Thanks JmsNxn, awesome stuff. I've reviewed it twice already and now I'm ready to closely review it line by line.
Just here to help, Daniel!
I may have fucked up some indices too 
, so just remember it's correct, I may have just fucked up some variable changes, lol. Too busy to double check every single number, but the main idea is absolutely true!!!I managed to iterate:
\[
\Delta^s f(z)\\
\]
That's in this paper from when I was an undergrad:
https://arxiv.org/abs/1503.06211
There are many more details on this Ramanujan \(\frac{d}{dz} \to \Delta\) correspondence.
