09/08/2007, 10:48 AM

Henryk -

I propose to introduce one sub-forum, called "integer tetration" or "natural tetration". "natural tetration" deals then with the more light-weight question of the tower of integral height, the tower of infinite height and the limits for the base parameter ... which seems so simple, that it even does not go further than for instance the wikipedia article (and we do not really need new articles about this)

But another aspect is then "infinite series of towers of integral height", including those of infinite height, - a subject, which seems not to be studied much but is interesting, as it is then a generalization of properties of known types of series, as I discussed this in my "infinite series of powertowers of like height - two conjectures" article; for instance the generalization for the result of the geometric series, which can be seem as series of towers of height 1 with increasing exponent x, as meant in {s,x}^^y in Andrew's collection of notations. I guess, there are some more interesting aspects, even in some encyclopedic, or basic referential, sense.

My very first discussion of infinite alternating series of s^^y, where s is a constant but y increases, (and s interestingly can exceed the bounds for towers of asymptotically infinite integral height) would then fit here too. (based on the eigensystem-properties it had eigenvalues of(1+log(t))^-1 , which are convergent sequences if log(t)>-1 and especially, if log(t)>1, which means t>e)

I'm also collecting results of infinite series of infinite series of towers of like height and like exponent x, but consecutive natural bases s, where then the Dirichlet-series are a special case (of height 1). I've nothing exiting here so far, but such a subforum would then be a good place to put the results into.

We could then also relocate these articles into this subforum and adapt their subject lines appropriately.

What do you think?

Gottfried

I propose to introduce one sub-forum, called "integer tetration" or "natural tetration". "natural tetration" deals then with the more light-weight question of the tower of integral height, the tower of infinite height and the limits for the base parameter ... which seems so simple, that it even does not go further than for instance the wikipedia article (and we do not really need new articles about this)

But another aspect is then "infinite series of towers of integral height", including those of infinite height, - a subject, which seems not to be studied much but is interesting, as it is then a generalization of properties of known types of series, as I discussed this in my "infinite series of powertowers of like height - two conjectures" article; for instance the generalization for the result of the geometric series, which can be seem as series of towers of height 1 with increasing exponent x, as meant in {s,x}^^y in Andrew's collection of notations. I guess, there are some more interesting aspects, even in some encyclopedic, or basic referential, sense.

My very first discussion of infinite alternating series of s^^y, where s is a constant but y increases, (and s interestingly can exceed the bounds for towers of asymptotically infinite integral height) would then fit here too. (based on the eigensystem-properties it had eigenvalues of(1+log(t))^-1 , which are convergent sequences if log(t)>-1 and especially, if log(t)>1, which means t>e)

I'm also collecting results of infinite series of infinite series of towers of like height and like exponent x, but consecutive natural bases s, where then the Dirichlet-series are a special case (of height 1). I've nothing exiting here so far, but such a subforum would then be a good place to put the results into.

We could then also relocate these articles into this subforum and adapt their subject lines appropriately.

What do you think?

Gottfried