wrong sign for value of Legendre polynomial at 0

[DaveWitteMorris]

As discussed in this sage-devel thread, legendre_P(n, 0) should be negative when n is congruent to 2 modulo 4, but sagemath returns a positive value:

sage: [legendre_P(n, 0) for n in range(0, 10, 2)]
[1, 1/2, 3/8, 5/16, 35/128]

The correct values are [1, -1/2, 3/8, -5/16, 35/128]. (The signs should alternate when restricted to even values of n.)

This is a pynac bug. It only arises in the code branch where n is an integer variable, so we get the correct values when n is real:

sage: [QQ(legendre_P(RR(n), 0)) for n in range(0, 10, 2)]
[1, -1/2, 3/8, -5/16, 35/128]

CC: @egourgoulhon

Component: symbolics

Keywords: orthogonal polynomial, legendre polynomial, pynac

Author: Dave Morris

Branch/Commit: 8131256

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/33962

[DaveWitteMorris]

Branch: public/33962

[DaveWitteMorris]

comment:2

Follow-up ticket: #33963. The patch does not solve the problem when n is a symbolic variable, rather than an explicit integer.

---

New commits:

c011409

trac 33962: fix legendre polynomial

[DaveWitteMorris]

Commit: c011409

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

comment:4

LGTM overall. Just a small change to the doc formatting:

-    TESTS::
+    TESTS:
 
-        # verify that :trac:`33962` is fixed
+    Verify that :trac:`33962` is fixed::
+
         sage: [legendre_P(n, 0) for n in range(-10, 10)]
         [0, 35/128, 0, -5/16, 0, 3/8, 0, -1/2, 0, 1,
-        1, 0, -1/2, 0, 3/8, 0, -5/16, 0, 35/128, 0]
+         1, 0, -1/2, 0, 3/8, 0, -5/16, 0, 35/128, 0]

Once changed, you can set a positive review on my behalf.

[sagetrac-git]

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

8131256

reviewer corrections

[sagetrac-git]

Changed commit from c011409 to 8131256

[DaveWitteMorris]

comment:6

Thanks for the review and the corrections!

I will set to positive review when the patchbot is green again.

[vbraun]

Changed branch from public/33962 to 8131256