Symbolic orthogonal and some other holonomic polynomials

[rwst]

At the moment just the Chebyshev polynomials are symbolic. Missing are hermite, laguerre, legendreP, legendreQ, ultraspherical (=gegenbauer), and, while the Stirling and Euler polynomials are not orthogonal, they are holonomic and belong to the standard repertoire, so we're including them here.

• symbolic Associated Legendre P(n,m,x) / Q(n,m,x) and P(n,x) / Q(n,x) polynomials / functions (symbolic Legendre / associated Legendre functions / polynomials #16813)
• symbolic Laguerre / associated Laguerre polynomials (symbolic Laguerre / associated Laguerre polynomials #17151)
• symbolic Gegenbauer polynomials (Upgrade to Pynac-0.6.5 #20475)
• symbolic Hermite polynomials (Upgrade to Pynac-0.6.5 #20475)
• symbolic Jacobi P polynomials (Jacobi P polynomials without pexpect-maxima #22326)
• symbolic Euler polynomials
• symbolic Stirling polynomials

Component: symbolics

Keywords: orthogonal, holonomic

Issue created by migration from https://trac.sagemath.org/ticket/16670

[rwst]

comment:2

For the record, using FLINT's arith_chebyshev_t_polynomial for chebyshev_T in ZZ[] is twice as slow as the current Sage implementation.

[fredrik-johansson]

comment:3

Really? For me flint constructs T_10000 in 0.009 seconds while Sage's chebyshev_T takes 1 second.

But evaluating a Chebyshev polynomial is a completely different problem. You want to do that by recurrence relations, not by constructing the polynomial and evaluating it.

[rwst]

comment:4

Ah, that's true. I only tested at T_500.

[rwst]

Branch: u/rws/make_all_orthogonal_polynomials_symbolic

[rwst]

Description changed:

--- 
+++ 
@@ -1 +1,3 @@
 At the moment just the Chebyshev polynomials are symbolic. Missing are `hermite`, `laguerre`, `legendreP`, `legendreQ`, `ultraspherical` (=`gegenbauer`), and, while the Stirling and Euler polynomials are not orthogonal, they are holonomic and belong to the standard repertoire, so we're including them here.
+
+* symbolic Associated Legendre P(n,m,x) / Q(n,m,x) and P(n,x) / Q(n,x) polynomials / functions (#16813)

[rwst]

Changed branch from u/rws/make_all_orthogonal_polynomials_symbolic to none

[rwst]

comment:6

The speed issue is now #16812

I think the scope of the ticket is too broad. Let's make it a meta-ticket.