symbolic Laguerre / associated Laguerre polynomials

[rwst]

Not only is laguerre(n,x) not symbolic, a naive implementation is already 2x as fast at n=100 and 10x at n=1000:

R.<x> = PolynomialRing(QQ, 'x')
def lag(n):
    return R([binomial(n,k)*(-1)^k/factorial(k) for k in xrange(n+1)])

• https://en.wikipedia.org/wiki/Laguerre_polynomials
• http://mathworld.wolfram.com/LaguerrePolynomial.html
• http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html

Depends on #17953

Component: symbolics

Keywords: special, function, holonomic, orthogonal

Author: Ralf Stephan

Branch/Commit: f0d809f

Reviewer: Marc Mezzarobba

Issue created by migration from https://trac.sagemath.org/ticket/17151

[rwst]

Branch: u/rws/symbolic_laguerre___associated_laguerre_polynomials

[rwst]

New commits:

6d10729	Simplify numerical evaluation of BuiltinFunctions
b6e1ed4	Merge remote-tracking branches 'origin/u/jdemeyer/ticket/17131' and 'origin/u/jdemeyer/ticket/17133' into ticket/17130
04b2b37	Merge branch 'u/jdemeyer/ticket/17130' of trac.sagemath.org:sage into t/17151/symbolic_laguerre___associated_laguerre_polynomials
3d3d1fa	17151: skeleton impl.
b3edc51	17151: implement symbolic Laguerre pol.
c8beac2	17151: implement laguerre(n,x)
1626a8b	17151: implement gen_laguerre

[rwst]

Commit: 1626a8b

[rwst]

Author: Ralf Stephan

[sagetrac-git]

Changed commit from 1626a8b to f727de5

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f727de5

17151: reduce evalf logic further; fix merge conflict

[kcrisman]

comment:6

For some reason this branch is red.

[sagetrac-git]

Changed commit from f727de5 to 6d05ac6

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

6d05ac6

Merge branch 'develop' into t/17151/symbolic_laguerre___associated_laguerre_polynomials

[rwst]

comment:9

sage -t --long src/sage/symbolic/expression.pyx  # 1 doctest failed
sage -t --long src/sage/matrix/matrix2.pyx  # 1 doctest failed
sage -t --long src/sage/graphs/bipartite_graph.py  # 1 doctest failed

[rwst]

Changed branch from u/rws/symbolic_laguerre___associated_laguerre_polynomials to u/rws/17151

[rwst]

comment:11

Fixed. Squashed it all into one commit.

EDIT: Or not.

---

New commits:

80f136b	Merge branch 'develop' into t/17151/symbolic_laguerre___associated_laguerre_polynomials
ea2ff9b	17151: symbolic Laguerre / associated Laguerre polynomials

[rwst]

Changed commit from 6d05ac6 to ea2ff9b

[mezzarobba]

comment:12

sage: laguerre(-9, 2.)
0.000000000000000

sage: laguerre(-9, 2).n()
1566.22186244286

[mezzarobba]

comment:27

Not a bug, but would be nice to have: the implementation doesn't seem to know that L(n, 0) = 1 and more generally L(n, α, 0) = binomial(n+α,n).

[rwst]

comment:29

Replying to @mezzarobba:

I also don't like that

sage: laguerre(3, polygen(QQ)).parent()
Symbolic Ring

Equivalently to #16813 this behaviour can only be changed when we have #18832. EDIT: pow is not a BuiltinFunction and chebyshev_T uses its own __call__ which is frowned upon. I would advise to not make this ticket dependent on the feature you suggest but to open a followup ticket.

[rwst]

Changed branch from u/rws/17151-1 to u/rws/17151-2

[rwst]

comment:31

With the new branch all other comments were addressed.

---

New commits:

6631c77

17151: symbolic Laguerre / associated Laguerre polynomials

[rwst]

Changed commit from 351b5c6 to 6631c77

[fchapoton]

comment:32

needs rebase, does not apply

[fchapoton]

Changed commit from 6631c77 to f0d809f

[fchapoton]

comment:33

rebased

---

New commits:

f0d809f

Merge branch 'u/rws/17151-2' into 6.9.b4

[fchapoton]

Changed branch from u/rws/17151-2 to public/ticket/17151

[mezzarobba]

comment:34

As explained in one of my previous comments, I still don't understand why you don't like the idea of using QQ['x'] to compute the coefficients in _pol_laguerre() (especially since #18282 has been merged and the problem with symbolic results being computed in Horner form no longer exists). But that's something that can be fixed later and this ticket has languished long enough already.

Thanks for working on it!

[mezzarobba]

Reviewer: Marc Mezzarobba

[rwst]

comment:35

Thanks for the review.

[vbraun]

Changed branch from public/ticket/17151 to f0d809f