# Tetration Forum

Full Version: Hooshmand's extension of tetration
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Quote:M.H. Hooshmand, August 2006, "Ultra power and ultra exponential functions", Integral Transforms and Special Functions, Vol. 17, No. 8, 549-558.

I recently found this reference in Wikipedia's Tetration article (which I moved to the talk page until it is clearly explained), but the full description of this extension was mainly contained in this article, both of which give the same reference. What bothers me is that these two pages describe this extension differently, and many of the uniqueness conditions are contradictory! Has anyone read this reference, and if so, then could they explain it in more detail? I would really like to know more about this extension, but unfortunately the Wikipedia article is lacking in clarity and completeness.

Andrew Robbins
Very important!
Is this method in connection with the implementation of a "continuous iteration" of the exp operator. Am I wrong? The Wikipedia article is not clear at-all. Nevertheless, I presume that his ultra-exponential and ultra-logarithm must exactly be what we call tetration and slog. We need to get the original article, asap!
GFR
First, it seems as if the article is in a refereed journal, and I don't know what that means. Second, the more I read about "uxp" in and of itself, the more I am sure it is just "linear" tetration (or what some other people call the "fractional part" extension of tetration). This is by no means new. If anything, the paper seems to focus on uniqueness theorems, so there might be something to be gained after all.

Andrew Robbins
andydude Wrote:If anything, the paper seems to focus on uniqueness theorems, so there might be something to be gained after all.

Hopefully it does not unveil as a condition that just favours the linear tetration. However thanks to our gold-digger Andydude!
I feel like I have come to a resolution to this issue on my part. The way that I interpreted the definition given in Wikipedia's UXP article is that there are 2 errors, which I will cover here.
1. The fourth condition requires that between (-1) and 0, UXP' is a:
"nondecreasing or nonincreasing" function, but this should read
"nondecreasing and nonincreasing", which means UXP' is a constant, which means UXP is a linear function.
2. The closed form given in the article defines UXP as:
$\text{uxp}_a(x) = \exp_a^{[x+1]}(\ (x)\ )$, but this should read
$\text{uxp}_a(x) = (\exp_a)^{\text{int}(x+1)}(\text{frac}(x+1))$, because of how "frac" is implemented on some CASs, and because this is much more clear than how it is described.

I hope the actual reference is better than this...

Andrew Robbins
Oh ... my God ! Let me ... sleep about that. GFR
Dear friend, you are wrong. If you look at the original paper (the main uniqueness theorem), then you find that the condition "nondecreasing or nonincreasing" is correct and is clearly different to the hypothesis " UXP' is a constant".

andydude Wrote:I feel like I have come to a resolution to this issue on my part. The way that I interpreted the definition given in Wikipedia's UXP article is that there are 2 errors, which I will cover here.
1. The fourth condition requires that between (-1) and 0, UXP' is a:
"nondecreasing or nonincreasing" function, but this should read
"nondecreasing and nonincreasing", which means UXP' is a constant, which means UXP is a linear function.
2. The closed form given in the article defines UXP as:
$\text{uxp}_a(x) = \exp_a^{[x+1]}(\ (x)\ )$, but this should read
$\text{uxp}_a(x) = (\exp_a)^{\text{int}(x+1)}(\text{frac}(x+1))$, because of how "frac" is implemented on some CASs, and because this is much more clear than how it is described.

I hope the actual reference is better than this...

Andrew Robbins
Hey Danesh, welcome at the forum.
Did you investigate Hooshmand's extension to some extent?
I wonder whether you clarify some more about it.
I didnt read the original article, but think its not an analytic extension, rather several times differentiable, is that true?
I also wonder how you found this forum and what your interests are.
Hi. Yes, it isn't analytic extension, but is unique extension. In fact the "ultra power" and "ultra exponential function" are gotten as the next step of the serial binary operations: addition, multiplication and power. I think you enjoy it if you get and read the original paper. You can see its abstract in the following site:

http://www.informaworld.com/smpp/content...order=page

I found this forum just by the google , while I was searching about the ultra exponential function, ultra power, tetration, etc.
Dear friends,

A new paper, related to ultra power and ultra exponential function (Hooshmand's extension of tetration), has been published in the journal
Integral Transforms and Special Functions. In it, another new function "Ultra power part function" and the dual of Uxp namely "Infra logarithm function (Iog)" ,that for a>1 is its inverse, have been introduced. You can see its abstract in