Add infinite q-Pochhammer symbol

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(z; q)_{\infty} has a nice expression for its coefficients in z. We implement this very important series as a lazy Laurent series.

We implement the Euler function (q; q)_{\infty} as a lazy Laurent series in q as well.

Depends on #32324

CC: @mantepse @fchapoton

Component: algebra

Keywords: q_pochhammer, euler function, LazyPowerSeries

Author: Travis Scrimshaw

Branch/Commit: 247faf7

Reviewer: Martin Rubey

Issue created by migration from https://trac.sagemath.org/ticket/34381

[tscrim]

Author: Travis Scrimshaw

[tscrim]

Commit: 80bc4e5

[tscrim]

comment:1

After doing this, we might want to move the construction of the underlying function to the lazy series ring. We might also want to special case composition of a lazy Laurent series with a monomial in a lazy Laurent series.

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New commits:

3f9045d	Implementing infinite q-Pochhammer symbol as a lazy Laurent series.
80bc4e5	Adding more documetnation and Euler's function.

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Branch: public/rings/infinite_q_pochhammer-34381

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Description changed:

--- 
+++ 
@@ -1 +1,3 @@
 `(z; q)_{\infty}` has a nice expression for its coefficients in `z`. We implement this very important series as a lazy Laurent series.
+
+We implement the Euler function `(q; q)_{\infty}` as well.

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Changed keywords from none to q_pochhammer, euler function

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Description changed:

--- 
+++ 
@@ -1,3 +1,3 @@
 `(z; q)_{\infty}` has a nice expression for its coefficients in `z`. We implement this very important series as a lazy Laurent series.
 
-We implement the Euler function `(q; q)_{\infty}` as well.
+We implement the Euler function `(q; q)_{\infty}` as a lazy Laurent series in `q` as well.

[tscrim]

comment:3

This also does not conflict with #32324.

[mantepse]

comment:4

Cool! It makes me happy to see so many possibilities opening up!

Question: what do you mean with moving the construction of the underlying function to the LazyLaurentSeriesRing?

Idea / Suggestion: maybe it makes sense to use "dummy" (or something similar) instead of "z" for the variable name, to make it clear to the reader that it plays no role. Of course, this would equally apply to all the other special functions. I am not sure about the benefit / cost ratio, I am not even sure that there would be a benefit.

[tscrim]

comment:5

Replying to @mantepse:

Question: what do you mean with moving the construction of the underlying function to the LazyLaurentSeriesRing?

For each of these special functions, we first construct a series f. So I propose having, e.g., L.sin() return f(z) = sin(z) in the variable of that lazy Laurent series ring. This avoids a composition call. I feel like it might make for a more natural home for, e.g., euler().

Idea / Suggestion: maybe it makes sense to use "dummy" (or something similar) instead of "z" for the variable name, to make it clear to the reader that it plays no role. Of course, this would equally apply to all the other special functions. I am not sure about the benefit / cost ratio, I am not even sure that there would be a benefit.

It gets substituted out every time. So it is never visible, and so I see no net change (although it would make a bug easier to spot if we ended up in the very unlikely scenario that the composition returned the wrong parent). If we wanted to do this, we should do it for all special functions simultaneously on a follow-up ticket.