A nice series for b^^h , base sqrt(2), by diagonalization

[Gottfried]

Hi Henryk -

just a short reply here.

Gottfried, we discussed that already in an much earlier post.
Your whole infinite matrix computation is just r e g u l a r i t e r a t i o n !!!

Well, I felt it is needed to make it explicitely, that it is special because of the (even blueish enhanced) hyperlink:

The matrix power approach makes use of the established method to obtain non-integer powers (and other analytic functions) of finite matrices via diagonalization.
This is applied to the truncations of the Carleman/Bell matrix.

where the keyword "finite matrices" occur. We had that several times and since I had assumed, that I had made it clear that I'm always working on infinite matrices, that remark is surprising.




Well, next question. "Why work with matrices if things are otherwise well known..." - I still don't claim something special. It's just my path into the matter: I came from a project, in which I compiled relations between pascal- Stirling, Euler- and other matrices operating on formal powerseries and stumbled on the possibility of iterating functions (other than that of addition, which I had already studied to some nice encounters with the zeta/eta-function)

Here I had the iteration of the exponential-function and as you might (frustrated? ) remember a hard and long time to even explain what I'm doing although Aldrovandi, Woon, Comtet and others were long time around in the scene. So its for personal reasons (experience with my matrix "toolbox"), possibly for "historical reasons" (to keep a track consistent) and recently for the connection of iteration-series with series of matrix-powers, which I've not seen yet except possibly in the form of the umbral-calculus, where it might be hidden behind the scene.
Just recently someone in sci.math asked for a set of symbolic matrix operations for mathematica or maple. Could be a nice start...

I've just reread Andrew's "exact entries for slog-operator" today and found a similar matrix-discussion there: it helps to understand since I've no training in functional analysis, the bit I had was in 1972 to 77 and only in relation for the computer-courses, which were my main subject.

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Well, but let's not lose the track here, for which I opended the thread. I think I'll have a much closer look at that series for tetration next time, to see whether and if, how, it's a special one.

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And, yes:

For me it seems you should do something healthy not always brood about the same things. Spring is coming, go out, have a look at the blue sky, or at the young girls passing

that's certainly what I should do. Hope I'll get things working/walking. And why the heck should the girls *pass*?
Ok - wish you all a good night

Gottfried