Transfinite sums related to a sequence

Question

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?

Answer 1

About questions 3 and 4:

3. It's known that for any countable ordinal $\alpha$ there is a subset of the reals with order type $\alpha$ under the usual order. Using this, given a countable ordinal $\alpha$ and a sequence of rationals $(x_\beta)_{\beta<\alpha}$ such that $((x_\beta)_{\beta<\alpha},<)$ has order type $\alpha$, if we set $z_\beta=\begin{cases}x_{\beta+1}-x_\beta\textrm{ if }\beta+1\le\alpha \\ \lim_{\gamma\rightarrow\beta}z_\gamma\textrm{ otherwise}\end{cases}$ for all $\beta<\alpha$, we have a convergent series of reals of length $\alpha$. (We can show the series converges to a real using transfinite induction up to $\alpha$ on $\beta$, at limit steps using the fact that sums commute with limits.)
4. The definition of sums for length-$\alpha$ series should work for any ordinal $\alpha$, countable or not, by transfinite recursion.