simplify_hypergeometric bug

[rwst]

The result of expansion of a 2F1 expression using simplify_hypergeometric differs from computing the numeric directly:

sage: hypergeometric((1/5, 4/5), (6/5,), 1/10*sqrt(5)*(sqrt(5) - 1))
hypergeometric((1/5, 4/5), (6/5,), 1/10*sqrt(5)*(sqrt(5) - 1))
sage: _.n()
1.04296605906833
sage: hypergeometric((1/5, 4/5), (6/5,), 1/10*sqrt(5)*(sqrt(5) - 1))
hypergeometric((1/5, 4/5), (6/5,), 1/10*sqrt(5)*(sqrt(5) - 1))
sage: _.simplify_hypergeometric()
0

Inside maxima:

maxima: hgfred([1/5, 4/5], [6/5], 1/10*sqrt(5)*(sqrt(5) - 1));
((1-(sqrt(5)-1)/(2*sqrt(5)))^(1/10)*assoc_legendre_p(-4/5,-1/5,1-(sqrt(5)-1)/sqrt(5))*gamma(1/5))/(5*((sqrt(5)-1)/(2*sqrt(5)))^(1/10))

CC: @slel

Component: calculus

Reviewer: Dima Pasechnik

Issue created by migration from https://trac.sagemath.org/ticket/22467

[rwst]

comment:1

User error. Assumptions given earlier interferred with simplify_hypergemetric.

[rwst]

comment:3

Hijacking ticket to prevent waste. Quite a similar error, this time for real.

[fchapoton]

comment:6

This is an issue in conversion from maxima:

sage: maxima('((1-(sqrt(5)-1)/(2*sqrt(5)))^(1/10)*assoc_legendre_p(-4/5,-1/5,1-(
....: sqrt(5)-1)/sqrt(5))*gamma(1/5))/(5*((sqrt(5)-1)/(2*sqrt(5)))^(1/10))')
((1-(sqrt(5)-1)/(2*sqrt(5)))^(1/10)*assoc_legendre_p(-4/5,-1/5,1-(sqrt(5)-1)/sqrt(5))*gamma(1/5))/(5*((sqrt(5)-1)/(2*sqrt(5)))^(1/10))
sage: bla=_
sage: bla.sage()
0

[fchapoton]

comment:7

Probably boils down to

sage: maxima('assoc_legendre_p(-4/5,-1/5,1)')
assoc_legendre_p(-4/5,-1/5,1)
sage: _.sage()
0

which in turn comes from

sage: gen_legendre_P(-4/5,-1/5,x)
0

[rwst]

comment:8

mpmath returns 0.000000000000000 with

        import mpmath
        from sage.libs.mpmath.all import call as mpcall
        mpcall(mpmath.legenp, -0.8, -0.2, 1)

and Wolfram Alpha agrees with it but also agrees with the nonzero value for the 2F1 expression so the simplification by Maxima seems wrong.

On another note there is this in Func_assoc_legendre_P._eval_special_values(self,n,m,x):

        if m > n:
            return ZZ(0)

However in Wikipedia that if |m|>l then P_l^m=0, so the code seems wrong (missing the abs).

[rwst]

comment:9

Confirmation from arb:

sage: CBF(1/10*sqrt(5)*(sqrt(5) - 1)).hypergeometric([1/5,4/5],[6/5])
[1.0429660590683 +/- 5.27e-14]
sage: CBF(1).legendre_P(-4/5,-1/5)
0

[fchapoton]

comment:10

Maybe because Legendre polynomials can be extended to rational parameters either by 0 or by hypergeometric functions. What we see looks like a clash of extensions, no ?

[rwst]

comment:11

No, the relevant extension is http://dlmf.nist.gov/14.3.E6 and it holds for x > 1 but the 2F1 argument here is 0.276 which makes x around 0.447, i.e. < 1. Agree?

[slel]

comment:12

Seems to have been fixed, maybe by #25034
(merged in Sage 9.3.rc5)?

In Sage 9.3 and above:

sage: a = hypergeometric((1/5, 4/5), (6/5,), 1/10*sqrt(5)*(sqrt(5) - 1))
sage: a.n()
1.04296605906833
sage: b = a.simplify_hypergeometric()
sage: b
2/5*5^(19/20)*(1/10)^(9/10)*(-1/10*sqrt(5)*(sqrt(5) - 1)
+ 1)^(1/10)*gamma(1/5)*gen_legendre_P(-4/5, -1/5,
-1/5*sqrt(5)*(sqrt(5) - 1) + 1)/(sqrt(5) - 1)^(1/10)
sage: b.n()
1.04296605906833

[dimpase]

Reviewer: Dima Pasechnik