slight generalisation of MZV auxiliary F-algebra

[fchapoton]

by allowing to start the odd indices at 1 or 3

📚 Description

The auxiliary F-algebra for Multiple Zeta Values is slightly generalised : its odd indices can start at 1 or 3.

This is motivated by the case of alternating MZV.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[tscrim]

Just a few little things.

[tscrim]

Thank you. Two more of the same doc change and one additional doc tweak.

[tscrim]

Suggested change

	The similar algebra on generators `f_1`, `f_3`, `f_5`, ... and `f_2`
	is now also available.
	The similar algebra with generators a subset of
	`f_1`, `f_3`, `f_5`, `\ldots` and `f_2` is now also available.

Perhaps this is closer to what you mean? The current addition makes it seem like it is just the same algebra as it is generated by the same set (unless I am missing something).

[fchapoton]

no, but I wrote something vague, for sure. There is a base ring which remains a polynomial in f_2, over which one has a shuffle algebra in one more generator f_1, in addition to the generators f_3, f_5, etc. Should I reformulate ?

[tscrim]

However, the code is saying something slightly different: you want to consider the algebras that have $f_2$ and $f_i, f_{i+2}, \ldots$ for some odd integer $i$. Of course, this includes $i = 1$ as per the ticket description, but I guess there is a slight mismatch between your intent and implementation? Perhaps it would be better to rename start (at least for the algebra input) to extended or some other boolean input to match this documentation?

[fchapoton]

ok, I have reformulated the doc so that any odd start index is allowed. Of course, only the start indices 1 and 3 are useful, but the implementation does not see any difference. One could even allow arbitrary subsets of odd integers, but this would just be re-doing the general implementation of shuffle algebras.

[tscrim]

Thank you. I have just made one more suggested micro-optimization if you want to include it. Once done or ignored, you can mark this as a positive review.

[fchapoton]

thanks a lot, Travis. I am setting to positive

[tscrim]

No problem.

[github-actions]

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