(adsbygoogle = window.adsbygoogle || []).push({}); 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 mathmatics 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

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

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