wrong sign for symbolic Legendre polynomial at 0

[gh-DaveWitteMorris]

Trac ticket #33962 fixed the sign of legendre_P(n, 0) when n is an explicit integer. The patch also tried to fix the sign when n is a symbolic integer variable, but the patch does not work. Even with the patch, we have:

sage: n = var("n")
sage: assume(n, "integer")
sage: assume(n, "even")
sage: legendre_P(n, 0)
2^(-n + 2)*gamma(n)/(n*gamma(1/2*n)^2)

This is missing a factor of (-1)^(n/2).

The incorrect value arises because pynac erroneously simplifies the expression pow(_ex_1, n / _ex2) to 1. (The constant _ex_1 is pynac's name for -1.) The problem seems to be that pynac thinks n / _ex2 is even: printing (n / _ex2).info(info_flags::even) at this point in the code yields 1 (meaning the flag is set).

[gh-DaveWitteMorris]

The bug is in case info_flags::even of pynac's mul::info method. A product is even if all of the factors are integers, and at least one of the factors is even. However, one of the factors in a mul object is the overall coefficient, and the method neglected to check that this coefficient is an integer.

In the example of the ticket description, pynac represents the expression n/2 as the product 1/2 * n, where 1/2 is the overall coefficient. This coefficient is obviously not an integer.

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New commits:

​c011409	trac 33962: fix legendre polynomial
​e29fdaf	trac 33963: fix legendre polynomial

[gh-DaveWitteMorris]

Follow-up: #33964. When n is odd, legendre_P(n, 0) should be 0.

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New commits:

​d967fb0

trac 33963: fix legendre polynomial

[tscrim]

LGTM modulo the doc change:

-        # verify that :trac:`33963` is fixed
+
+    Verify that :trac:`33963` is fixed::
+
         sage: n = var("n")

[tscrim]

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

[gh-DaveWitteMorris]

Thanks for the review and corrections of this ticket, too.

I rebased on #33962, and will set to positive review when the patchbot is green again.

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New commits:

​8131256	reviewer corrections
​a3608a4	trac 33963: fix legendre polynomial