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.

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.

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.