B(1) = +½ set 9: Appell sequences

[Parcly-Taxel]

References to other Issues or PRs

Part 9 of the changes formerly included in #23926, but initiated because of a comment on #23989 suggesting a move of the combinatorial polynomial generators to the polys module.

Brief description of what is fixed or changed

The polynomial sequences associated with bernoulli(), genocchi() and euler() are all Appell sequences. This means they can be generated efficiently by passing different parameters to the same piece of code, something Luschny hints at when he discusses the so-called central Bernoulli and Swiss-knife polynomials, the latter renamed here to André polynomials.

This PR implements the five specific Appell sequences mentioned in the previous paragraph.

Release Notes

• polys
  • New appellseqs module generating Appell sequences: Bernoulli, central Bernoulli, Genocchi, Euler and André polynomials.

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• polys
  • New appellseqs module generating Appell sequences: Bernoulli, central Bernoulli, Genocchi, Euler and André polynomials. (#24014 by @Parcly-Taxel)

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#### References to other Issues or PRs

Part 9 of the changes formerly included in #23926, but initiated because of [a comment](https://github.com/sympy/sympy/pull/23989#discussion_r957861664) on #23989 suggesting a move of the combinatorial polynomial generators to the `polys` module.

#### Brief description of what is fixed or changed

The polynomial sequences associated with `bernoulli()`, `genocchi()` and `euler()` are all [Appell sequences](https://en.wikipedia.org/wiki/Appell_sequence). This means they can be generated efficiently by passing different parameters to the same piece of code, something Luschny hints at when he discusses the so-called _central Bernoulli_ and _Swiss-knife polynomials_, the latter renamed here to _André polynomials_.

This PR implements the five specific Appell sequences mentioned in the previous paragraph.

#### Release Notes

<!-- BEGIN RELEASE NOTES -->
* polys
    * New `appellseqs` module generating Appell sequences: Bernoulli, central Bernoulli, Genocchi, Euler and André polynomials.
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[sympy-bot]

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Hi, I am the SymPy bot (v167). I've noticed that some of your commits add or delete files. Since this is sometimes done unintentionally, I wanted to alert you about it.

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The following commits add new files:

• e875bdb:
  • sympy/polys/appellseqs.py
• 840c632:
  • sympy/polys/tests/test_appellseqs.py

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[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

       before           after         ratio
     [41d90958]       [5f0b7d3b]
     <sympy-1.11.1^0>                 
-         963±3μs          636±2μs     0.66  solve.TimeSparseSystem.time_linear_eq_to_matrix(10)
-     2.80±0.01ms         1.19±0ms     0.43  solve.TimeSparseSystem.time_linear_eq_to_matrix(20)
-     5.61±0.02ms         1.75±0ms     0.31  solve.TimeSparseSystem.time_linear_eq_to_matrix(30)

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[oscarbenjamin]

polys

• New appellseqs module generating Appell sequences, including the Bernoulli and Euler polynomials.

The release note should list the new public functions.

[oscarbenjamin]

I left a couple of comments but otherwise I think this looks good.

I notice that the line spacing in orthopolys is inconsistent with sometimes one and sometimes two blank lines between functions. Broadly throughout SymPy the convention is two blank lines between top level definitions but it seems that in polys the convention is more often one line so it's probably better to stick with that.

[oscarbenjamin]

Looks good.