(07/28/2022, 08:12 AM)Daniel Wrote: Yes, you get where I'm going with this. Ironically, I have always received more support from physicist than mathematicians. Part of my interest in a Taylor's series approach to iterated functions and tetration is that complex numbers and even matrices can be plugged in. Maybe even infinite matrices, but proving their convergence is beyond me. One of my favorite papers is what I believe to be the first paper devoted to general fractional(continuous) iteration:
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps
J. Math. Phys. 39, 5324 (199
It makes use of infinite Bell matrices which are a variation of Carleman matrices.
Ironically, in the same vein as you're talking, most of the mathematicians who I've talked to and given me the time of day are analytic number theorists. This somehow related to how I performed mellin transforms and blah blah blah. Iterating a super function relationship is far more interesting. No one cares about that though. So I made a couple of cool observations at u of t involving analytic number theory--but all I cared about was tetration/iteration/recursion.
I got some love from physicist's at u of t too, but that was mostly transferred from the number theory guys. The way I manipulated number theory functions, was applicable to schrodinger solutions.
Nothing to do with tetration. So I don't give a fuck. And I didn't give a fuck.
But if we're talking background. You say physics, and what I'm presuming is more classical (not quantum mechanics). I have a good background in analytic number theory; and oddly enough--it transfers to tetration.
I bet if we work together we'll all make leaps in tetration.
EDIT:
It's also bugging me that I screwed up the above equation. It should be:
\[
\Gamma(1-z)A^{z-1}x = \int_0^\infty e^{-At}xt^{-z}\,dt\\
\]
I forgot a dumb negative sign.
Either way, this stuff appears all the time in advance number theory stuff....