03/30/2008, 09:24 AM

Gottfried Wrote:...

y = x{4,b}h for iterated exponentiation beginning at x: b^...^b^b^x

(and from earlier discussions)

y = x{3,b}h for iterated multiplication beginning at x: x*b^h

y = x{2,b}h for iterated addition beginning at x: x+b*h

for my needs for the time being,

the "height"-function

h = hgh(x,b)

if

x = 1 {4,b} h

= b [4] h // related to the tetrational notation

...

I personally think that the Arrow-Iteration-Section notations discussed in my first post cover most of these use cases, but U-tetration is different enough to require a special notation. Here are my recommendations:

but I've seen other notations elsewhere. The one I've seen used the most is x^^y@a, although I had also used y`x`a in the past. Also, GFR uses x$y*a or something like that, which I find confusing. Thats all about iter-exp.

Starting from scratch using Arrow-Iteration-Section notation, we find that the natural expression in ASCII is (x^)^y(a) which could be shortened to x^^y(a) which means the corresponding notation for iterated decremented exponentials is (x^-)^y(a) which could be shortened to x^-^y(a), what do you think? About iter-dec-exp/U-tetration, this would mean that your "height" function is h = hgh(x, b, a) = b^-^\x(a) and h = hgh(x, b) = b^-^\x which I would've called the "super-decremented-logarithm" or something.

We might even go so far as to use similar notations for superroot and superlog, so srt_n = (/^^n) and slog_b = (b^^\).

While I'm at it, I might as well summarize the other suggestions (based on BO's):

I must say, the slash notation is by far the most expressive tetration notation I've ever seen. It allows full expression of practically anything I can think of that is hyperop/tetration related. As you can see, it covers many topics that do not have a specialized notation yet.

Andrew Robbins