I developing a theory of a general shift function sequence transform

[bqllpd]

I need help developing a theory of a general "shift function sequence transform"

I'm new here and this is my first post. In 1999 I read part of "The Book of Numbers" by John Conway and Richard Guy and came across a section titled "Jackson's difference fans". In their book, there's no mention of the math that decribes this fanning process, so over the past few years, two friends of mine and I developed the mathematics for it. For some reason, they didn't want their name on the paper and I'm not really getting help with it anymore, but I'm now trying to discover a general method for describing any sequence transform by using what I call a "shift function sequence transform".

Lets's see if this $\LaTeX$ posts right...
Let $\textbf I$ denote the Identity operator so that $\textbf Ia_n=a_n$ where $a_n$ is a complex number and $n$ is an integer. Let $\textbf E$ denote the shift operator such that $\textbf Ea_n=a_n+1$ and $\textbf E^ka_n=a_{n+k}$

Define $f(\textbf E)a_n$ be a shift function of the elements of the sequence. E few examples are mentioned in another group I belong to in yahoo called Math for Fun and the post I wrote a few days ago can be found at http://groups.yahoo.com/group.mathforfun

I'm sorry I had to post an outside group here. I just don't have enough time to retype everything because I have an hour time limit each day.

If $a_n$ is the original sequence and $f(\textbf E)a_n=b_n$, then $f(\textbf E)b_n=f^2(\textbf E)a_n$ If this is continued, then the $k$-th shift function of the sequence is $f^k(\textbf E)a_n$ Now define $Ta_n$ as the first elements of each new sequence produced by $f^k(\textbf E)$ for each $k$ so that $Ta_n=f^n(\textbf E)a_0$

This is where I start to have problems. I just don't know what the notation would be in general if this same "fanning" process is repeated. I know what it is if $f(\textbf E)=\textbf E-\textbf I$ and $f(\textbf E)=\textbf E+\textbf I$ as well as $f(\textbf E)=\textbf E/\textbf I$ and $f(\textbf E)=\textbf E*\textbf I$ but I don't know a general rule for any shift function and would appreciate it if anyone can help.

Regards,
Jason

 

[rbj]

Jason, to do $$\LaTeX$$ here, you have to use the [ tex ] and [ /tex ] markup constructs. the dollar signs don't do zilch.

 

[tacman]

Dear Jason,

Thank you for sharing your work and your interest in developing a theory of a general "shift function sequence transform". It sounds like you have put a lot of effort into understanding and exploring this concept.

One suggestion I have is to try to break down the problem into smaller, more manageable pieces. It may be helpful to start by defining the general shift function sequence transform in a more precise way. What are the inputs and outputs of this transform? Can you provide a formal definition or mathematical expression for it?

Once you have a clear definition, you can then start exploring different properties and characteristics of the transform. This may involve looking at specific examples, as you have already done, and trying to identify patterns or general rules that apply to all cases.

Additionally, it may be helpful to seek out other resources or experts in the field who have studied similar concepts. They may be able to offer insights and suggestions that can further your understanding and development of the theory.

Keep up the good work and don't be discouraged if progress is slow. Developing a theory takes time and effort, but it can also be a rewarding and exciting journey. Best of luck to you!