# Tetration Forum

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Hello guys (and girls if there are any),

here is a very first version of the FAQ.
It is more meant as a crystallisation seed than anything else.
But perhaps it can be already quite useful for the one or another.
I was wondering, since we are accumulating a lot of standards, tables, formulas, equations, and so on, whether we should collectively make a Mathematical Tables and Formulae like document? It would be so that it gives you just what you need since you already learned it before. Not an overly verbose introduction, just a reference for those formulas that you tend to forget after while... It certainly should not include new ideas or theories and the like, or open questions. Only things that have been around a while like the infinite exponential or Bell matrices, or things that have been proven.

I have such a document started, and anyone who would like to suggest or recommend parts or contribute to it, would be most welcome, and we will all be credited of course. I have two formats for viewing:
Tables and Formulae - TeX source
Tables and Formulae - PDF output
Let me know what you think so far, what order you think sections should be in, and whether or not you think this is a good idea.

Andrew Robbins
By the way, have we found a suitable term for the tetrational equivalent of an exponent? For example, given $2^{3}$, 2 is the base, and 3 is the exponent. So, given ${}^{3} 2$, 2 is the base, and 3 is the...?

I tried using tetrant (or tetrent), but it doesn't quite sound right when said out loud...

On a related note, a specific iteration could be called an iterate (the -ate sounding like "it", not like "ate"). Extending this linguistic idea to tetration, could we say that ${}^{4} 5$ is the fourth tetrate of five?
jaydfox Wrote:For example, given $2^{3}$, 2 is the base, and 3 is the exponent. So, given ${}^{3} 2$, 2 is the base, and 3 is the...?
The hyper-exponent. This also works for all hyper-operators, not just tetration.

jaydfox Wrote:could we say that ${}^{4} 5$ is the fourth tetrate of five?
Sounds ok, but according to GFR's recommendation, that would be the 4th tower of 5.
The pronunciation I like the most, though, is 5-tetra-4.

On another note, have you read my handout from Tetration in Context? (also mentioned here) in which I talk a little bit about that. Ignore the "hyper-power" terminology, as I have replaced this in my head with GFR's less-of-a-misnomer: "tower".

Andrew Robbins
Are you talking about replacing the nth tower terminology with the nth tetrate term? Because if that's so, then we could use the term tower for nested exponentials. So its a matter of the following options:
• The nth (homogeneous) tower is ${}^{n}b$ and a nested exponential is $a^{b^{c^{\cdot^{\cdot}}}}$.
• The nth tetrate is ${}^{n}b$, and a (heterogeneous) tower is $a^{b^{c^{\cdot^{\cdot}}}}$.

Anyone care to vote?

Andrew Robbins
andydude Wrote:I was wondering, since we are accumulating a lot of standards, tables, formulas, equations, and so on, whether we should collectively make a Mathematical Tables and Formulae like document? It would be so that it gives you just what you need since you already learned it before. Not an overly verbose introduction, just a reference for those formulas that you tend to forget after while... It certainly should not include new ideas or theories and the like, or open questions. Only things that have been around a while like the infinite exponential or Bell matrices, or things that have been proven.

Andrew Robbins
Very good idea! I noticed that need myself some weeks ago, and also Jay asked smething like this, as I understood his post. I think it is high time to reassure, whether and if, where we are converging with our approaches. A formula-notebook is a good instrument to have the "look back".
I'll see, what I can contribute to this notebook. Yours is a promising start.

Gottfried
andydude Wrote:Are you talking about replacing the nth tower terminology with the nth tetrate term? Because if that's so, then we could use the term tower for nested exponentials. So its a matter of the following options:
• The nth (homogeneous) tower is ${}^{n}b$ and a nested exponential is $a^{b^{c^{\cdot^{\cdot}}}}$.
• The nth tetrate is ${}^{n}b$, and a (heterogeneous) tower is $a^{b^{c^{\cdot^{\cdot}}}}$.

Anyone care to vote?

Andrew Robbins

a) "Tower of height n", n'th tetrate (the latter, if context of continuous operation is focused), heterogeneous, leave "nested" for some more special cases (though I have no idea actually)

b) @Jay: top exponent, initial exponent; preferring "initial" to connect to iteration-theory("initial state") and due to the opportunity to be able to better talk about infinite towers and their fixpoints then.

Gottfried
Gottfried Wrote:a) "Tower of height n", n'th tetrate (the latter, if context of continuous operation is focused), heterogeneous, leave "nested" for some more special cases (though I have no idea actually)
There are only two kinds of cases with heterogeneous towers / nested exponentials: $a^{b^c}$ as an expression, and $x \rightarrow a^{b^x}$ as a function, there are no other special cases. Even if the x is somewhere else, you can view it as the composition of nested exponential functions and other functions. Even the expression ${e^{ae^{be^c}}} = {(e^a)}^{({e^b)}^{(e^c)}}$ is just a tower (of hight 3 or 4) with weird bases. Reminds me of one paper that tried to call these expressions binary iterated powers but they're still towers to me.

Gottfried Wrote:b) @Jay: top exponent, initial exponent; preferring "initial" to connect to iteration-theory("initial state") and due to the opportunity to be able to better talk about infinite towers and their fixpoints then.
I've actually noticed that there is a problem with this. Lets say you have a tower of even hight n, and you want to consider a base/exponent at tower level m=n/2. For all levels > m, level m is a base. For all levels < m, level m is an exponent. So in general what can these be called, if not bases, and not exponents? I like to call them elements or tower elements, but thats just my vote. This comes up so rarely, that it may not make much of a difference. I like the idea of calling them all exponents, but just remember, they are also bases. Also, the levels have been called tower heights, if i remember right.

Andrew Robbins
@Gottfried
Oh, about your top-down towers as opposed to bottom up towers, I've put some thought into those as well, and I've come up with a notation for those based on Barrow/Shell's tower notation.

${\overset{k=1}{\underset{n}{\text{\huge \rm T}}}}\, a_k := {\underset{k=1}{\overset{n}{\text{\huge \rm T}}}}\, {a}_{n-k+1} = {a}_{n}^{{a}_{n-1}^{\cdot^{\cdot^{a_2^{a_1}}}}}$

This way of looking at it makes it not so much a matter of notation (as yours is) but a matter of indexing. The reversed indexing allows you to take the limit in the other direction as I think you were talking about. Also using this notation, your downward towers can be written:

$\lim_{n \rightarrow \infty} \, \overset{k=1}{\underset{n}{\text{\huge \rm T}}}\, (a_k; z) = {}_{\cdot^\cdot}{a_2^{a_1^z}}$

My only worry with this kind of tower is that the limits will be found to always evaluate to ${}^{\infty}\left(\lim_{n\rightarrow\infty} a_n\right)$ and will not provide a very "rich" convergence landscape.

Andrew Robbins

PS. I also find it interesting that the "k=1" at the top of the big-T fits on the line of the T which is bigger and the "n" fits at the bottom of the T which is smaller, so the top-down notation fits better with the T notation, just as the top-down view of nested exponentials is how they are actually evaluated, bringing us closer to reality.
Yes, great idea!
I would like to see formulas for regular iteration at an arbitrary fixed point (power series coefficients and iteration formula). I can provide them at a later time.

I would also emphasize not to use too much invented notation.
For example the one base towers are expressible by $\exp_b^{\circ n}$.
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