Trac #32964: very slow doctest in functions.orthogonal_polys

{{{
$ sage -t --warn-long 10.0 --random-
seed=80495217684824677825915603387163671014
src/sage/functions/orthogonal_polys.py
**********************************************************************
File "src/sage/functions/orthogonal_polys.py", line 2084, in
sage.functions.orthogonal_polys.Func_ultraspherical
Warning, slow doctest:
    for N in range(100):
        n = ZZ.random_element(5, 5001)
        a = QQ.random_element().abs() + 5
        assert ((n+1)*ultraspherical(n+1,a,x) -
2*x*(n+a)*ultraspherical(n,a,x) +
(n+2*a-1)*ultraspherical(n-1,a,x)).expand().is_zero()
Test ran for 261.17 s
    [400 tests, 273.87 s]
}}}

Since the test is randomized, we don't need to repeat it a hundred
times. Coverage is provided more by the number of testers. And we
shouldn't use `assert()` in EXAMPLES...

URL: https://trac.sagemath.org/32964
Reported by: mjo
Ticket author(s): Michael Orlitzky
Reviewer(s): Frédéric Chapoton

Author: Release Manager

src/sage/functions/orthogonal_polys.py

@@ -498,8 +498,7 @@ def _eval_(self, n, x):
             sage: chebyshev_T(5,Qp(3)(2))
             2 + 3^2 + 3^3 + 3^4 + 3^5 + O(3^20)
             sage: chebyshev_T(100001/2, 2)
-            doctest:...: RuntimeWarning: mpmath failed, keeping expression unevaluated
-            chebyshev_T(100001/2, 2)
+            ...chebyshev_T(100001/2, 2)
             sage: chebyshev_U._eval_(1.5, Mod(8,9)) is None
             True
         """
@@ -1914,7 +1913,7 @@ class Func_hermite(GinacFunction):
         ...
         RuntimeError: hermite_eval: The index n must be a nonnegative integer
 
-        sage: _ = var('m x')
+        sage: m,x = SR.var('m,x')
         sage: hermite(m, x).diff(m)
         Traceback (most recent call last):
         ...
@@ -1965,7 +1964,7 @@ def __init__(self):
 
         EXAMPLES::
 
-            sage: _ = var('n a b x')
+            sage: n,a,b,x = SR.var('n,a,b,x')
             sage: loads(dumps(jacobi_P))
             jacobi_P
             sage: jacobi_P(n, a, b, x, hold=True)._sympy_()
@@ -1979,7 +1978,7 @@ def _eval_(self, n, a, b, x):
         """
         EXAMPLES::
 
-            sage: _ = var('n a b x')
+            sage: n,a,b,x = SR.var('n,a,b,x')
             sage: jacobi_P(1,n,n,n)
             (n + 1)*n
             sage: jacobi_P(2,n,n,n)
@@ -2083,17 +2082,20 @@ class Func_ultraspherical(GinacFunction):
         sage: t = PolynomialRing(RationalField(),"t").gen()
         sage: gegenbauer(3,2,t)
         32*t^3 - 12*t
-        sage: _ = var('x')
-        sage: for N in range(100):
-        ....:     n = ZZ.random_element(5, 5001)
-        ....:     a = QQ.random_element().abs() + 5
-        ....:     assert ((n+1)*ultraspherical(n+1,a,x) - 2*x*(n+a)*ultraspherical(n,a,x) + (n+2*a-1)*ultraspherical(n-1,a,x)).expand().is_zero()
+        sage: x = SR.var('x')
+        sage: n = ZZ.random_element(5, 5001)
+        sage: a = QQ.random_element().abs() + 5
+        sage: s = (  (n+1)*ultraspherical(n+1,a,x)
+        ....:      - 2*x*(n+a)*ultraspherical(n,a,x)
+        ....:      + (n+2*a-1)*ultraspherical(n-1,a,x) )
+        sage: s.expand().is_zero()
+        True
         sage: ultraspherical(5,9/10,3.1416)
         6949.55439044240
         sage: ultraspherical(5,9/10,RealField(100)(pi))
         6949.4695419382702451843080687
 
-        sage: _ = var('a n')
+        sage: a,n = SR.var('a,n')
         sage: gegenbauer(2,a,x)
         2*(a + 1)*a*x^2 - a
         sage: gegenbauer(3,a,x)