A Notation Question (raising the highest value in pow-tower to a different power)

[Micah]

Dear users of the Tetration Forum, I am a new member and I have a question on notation for a function I have been trying to investigate.  I would like to denote a function using tetration notation, where the iteration index k is integral and the highest tetrand (proper use?) is exponentiated to a different real value -(r) for positive real r.  I have been denoting it with the following, but am sure that this notation doesn't exactly mean what I want, I am hoping that someone may have a proper analytic formula to represent this function.  

Code:

[tex] $$f(x \vert k, r) = (x \uparrow \uparrow k)^{\uparrow(-r)} = x^{x^{x^{\rddots{x^{-r}}}}}$$ [\tex]

where the \rddots is defined (in preamble) as: \def\rddots#1{\cdot^{\cdot^{\cdot^{#1}}}}

This function has some interesting properties I would like to explore, but I want to make sure that I am using consistent notation, I hope that I have made clear what I mean by this, if I have not please feel free to message me privately, or respond to the thread in general. Thankyou very much for your time and consideration. 

PS - I apologize if I am breaking a forum rule by posting this in any way (I hope you will show mercy as I am new).  I am hoping to discover some of the properties of the above function and may have too hastily posted, please forgive my rudeness, ignorance of notation, and informality for posting LaTeX code in this way as I am not sure how to properly format my latex expressions, I tried including an image, but it seems as though you may only embed images through links, I may look at hosting images of LaTeX formatted equations in future posts via image sharing cites like imgur etc... (is this how it is done here?) I have also included a png of the formula.

[jaydfox]

Welcome, Micah!

The forum software does support a subset of the TeX language.  It's not the full TeX language, but it's enough to do a reasonable amount of math.

As an example, a generic Taylor series:

\( f(x-c) = \sum_{k=0}^{\infty}\[a_k (x-c)^{k}\] \)

Code:

[tex]f(x-c) = \sum_{k=0}^{\infty}\[a_k (x-c)^{k}\][/tex]

I'm not an expert at TeX, so I'm sure even my simple example could be improved upon.  I just wanted to let you know that the forum does support a subset of TeX.

[Micah]

Welcome, Micah!

The forum software does support a subset of the TeX language.  It's not the full TeX language, but it's enough to do a reasonable amount of math.

As an example, a generic Taylor series:

\( f(x-c) = \sum_{k=0}^{\infty}\[a_k (x-c)^{k}\] \)

Code:

[tex]f(x-c) = \sum_{k=0}^{\infty}\[a_k (x-c)^{k}\][/tex]

I'm not an expert at TeX, so I'm sure even my simple example could be improved upon.  I just wanted to let you know that the forum does support a subset of TeX.

Hey Jay, 

     Thanks for the quick response, I was trying to get the notation working, does it have to be within a <code> environment?  I'm sorry, I'm really a novice when it comes to posting on these mathematical forums, Its always beautiful to see new expressions of Taylor's series, thankyou for that!  Any thoughts on the expression of the function in this thread?  What is the standard operator for hyper-4 in this forum? I am used to using the up-arrow notation, is that received okay here? 

Thanks so much! 

-Micah

[jaydfox]

Hey Jay, 

     Thanks for the quick response, I was trying to get the notation working, does it have to be within a <code> environment?  I'm sorry, I'm really a novice when it comes to posting on these mathematical forums, Its always beautiful to see new expressions of Taylor's series, thankyou for that!  Any thoughts on the expression of the function in this thread?  What is the standard operator for hyper-4 in this forum? I am used to using the up-arrow notation, is that received okay here? 

Thanks so much! 

-Micah

You were really close to the correct format in your first post.  The closing tag of the tex block needs to be closed with a forward slash, not a backwards slash.  No need to put it into a code block.

Another way you can see how it works is to reply to my first post, then click the View source button.  It's the last button on the toolbar when you are in the post editor.  I'll post a screenshot of that below.  When you push that button while you are replying, it will convert the original post to BB code.  I always use the BB code.  I don't like using the wysiwig editor.

   

[jaydfox]

Any thoughts on the expression of the function in this thread?  What is the standard operator for hyper-4 in this forum? I am used to using the up-arrow notation, is that received okay here?

I don't think there's a standard notation.  We try to be flexible, as long as it's clear what you mean.  When in doubt, define your notation before you use it, or after the first use, and then just be consistent.

As far as notations we use here, I've seen the use of the double caret, such as 4^^3, which would be 4^(4^4), assuming that the single caret is exponentiation.  This is essentially the same as up-arrow notation you mentioned (Knuth arrow notation?).

I myself am partial to the left-sided superscript:

\( {}^{3}{4} = 4^{4^{4}} \)

I can't remember if there's a better way to write a left-sided superscript in TeX.  I haven't been on the forum much for the past 4-5 years.  Maybe one of the other regular forum members will remember.

The downside of the left-sided superscript is that it doesn't have a clear extension to pentation.  With the up-arrow notation, pentation would be a triple up arrow.  So when you're getting into a deep discussion about iteration theory, up-arrow notation is probably more appropriate.  If you're getting into a deep discussion about tetration specifically, then the left-sided superscript is a little more compact.  But it's all a question of preference.

[jaydfox]

There is actually another notation you'll see some of use, when discussing iteration theory, Abel functions, etc.

\( \exp^{k}(x) \)

Here, k is taken to be the number of iterations of the exponential function.  I believe the mathematical term is "functional iteration".  This format is very easily confused with exponentiation.  For example, in trigonometry, you'll often see the sine and cosine functions raised to integer powers.  For example, the triple angle formula for the sine function is:

\( \sin{{3}{\theta}}={3}\sin{\theta} - {4}\sin^{3}{\theta} \)

Here, the sin^3 means to find the sine of theta, then raise that value to the third power, i.e., exponentiate:
\( \sin^{3}{\theta} = \(\sin{\theta}\)^{3} \)

However, with the functional iteration notation, k is taken to be the number of iterations:
\( \exp^{3}(x) = \exp\(\exp\(\exp(x)\)\) \)

This notation works very well for integer iterations, but it becomes poorly defined for non-integer values of k.  Negative values of k would imply iterations of the inverse of exponentiation, i.e., logarithms.

\( \exp^{-3}(x) = \log\(\log\(\log(x)\)\) \)

[Micah]

\( \exp^{3}(x) = \exp\(\exp\(\exp(x)\)\) \)

Using the notation that you discuss here, I believe that the function I am curious about may be represented more elegantly by using the notation for it shown below: 

\( f(z,k,r) = \exp_z^k(-r) \) 

Here the z subscript denotes that the base of the exponentiation in this case is the real valued z.  If I were to expand the above function would the notation equate to the following? 

\( f(z=2,k=2,r=1) = 2^{2^{(-1)}} \) 

Thanks for the discussion, and advice on inserting TeX properly. 

-Micah

[sheldonison]

...
\( f(z,k,r) = \exp_z^k(-r) \) 

Here the z subscript denotes that the base of the exponentiation in this case is the real valued z.  If I were to expand the above function would the notation equate to the following? 

\( f(z=2,k=2,r=1) = 2^{2^{(-1)}} \) 
...

The notation I like best for clarity to distinguish from raising powers is something like this; with an optional "\circ" and  with brackets around the iteration count of k.
\( f(z,k,r) = \exp_z^{[\circ k]}(-r) \)

[Micah]

The notation I like best for clarity to distinguish from raising powers is something like this; with an optional "\circ" and  with brackets around the iteration count of k.
\( f(z,k,r) = \exp_z^{[\circ k]}(-r) \)

The open circle notation is excellent thanks for the tip.
   -Micah