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Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products of all $k+1$ tuples of elements taken from {$S_\beta| \beta < \alpha$} if the sum exists. Also, define $(S+k)$ for $k \in \mathbb{N}$ to be the sequence $(S+k)_n = S_{k+n}$, essentially "dropping" the terms up to index $k$.

For example, $S_\omega$ is the standard sum of a series from Calculus, $S_{\omega+1} = \sum_{n=1}(S_n \cdot \sum_{k=n+1}^\infty S_k)$, and in general $S_{\alpha+1} = \sum_{n=1}S_n (S+n)_\alpha$.

I've managed to convince myself it has a few self-consistent properties:

  • If $S_\alpha$ exists, then $(S+k)_\alpha$ exists.
  • If $S_\alpha$ exists and $\beta < \alpha$, then $S_\beta$ exists.

My questions are:

  1. Has this concept been studied before, and if so can you provide references? I'm more concerned about the formal properties of these sequences, such as the algebra it generates. I suspect there's some paper or book on Generating Functions that would cover it.

  2. Given a set S of Complex numbers and an ordinal $\alpha$, you can create a new set $S_\alpha$ containing $s_\beta$ for every $\beta <= \alpha$ and every sequence $s$ of numbers taken from $S$. Is there a paper that gives more detail on this operation? Interesting choices for $S$ might be the set of all roots of unity, or the set of zeros from some analytic function.

  3. Except for the constant 0 sequence, is it true that if you repeatedly transfinitely extend a sequence you will find a large enough ordinal where the sum does not exist? If so, what is the largest ordinal(or the supremum of all such ordinals) possible to extend a sequence to?

  4. My definition should work for countable ordinals, but I'm not entirely sure it's well-defined for anything larger. Is there a more natural/general definition I should be using that captures the concept better?

Thank you,

-- Michael Burge

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In question 2, if you really want $S$ to be a set of real numbers, then the set of all roots of unity is just $\lbrace1,-1\rbrace$, not likely to be an interesting choice. – Gerry Myerson Nov 17 '10 at 1:57
You're right; changed to Complex Numbers, but the underlying set doesn't matter too much. – Michael Burge Nov 17 '10 at 2:56
Maybe I'm old fashioned, but I don't much care for definitions of sequences that treat limit and successor ordinals this differently. – Theo Johnson-Freyd Nov 17 '10 at 2:59
Can you do an example or two for us? Say $S_n = 2^{-n}$ or something? – Gerald Edgar Nov 17 '10 at 14:41

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