add explict doctest and additional example to AsymptoticRing.coeffici…

…ents_of_generating_function

src/sage/groups/misc_gps/argument_groups.py

@@ -1461,6 +1461,15 @@ def __pow__(self, exponent):
             1
             sage: S(-1)^3  # indirect doctest
             -1
+
+        Check that the results may live in other parents too::
+
+            sage: x = SR.var('x')
+            sage: elem = S(-1)^x; elem  # indirect doctest
+            (-1)^x
+            sage: elem.parent()
+            Symbolic Ring
+
         """
         result = self._element_ ** exponent
         P = self.parent()

src/sage/rings/asymptotic/asymptotic_ring.py

@@ -4415,6 +4415,16 @@ def coefficients_of_generating_function(self, function, singularities, precision
             ....:    'n', precision=5))
             True
 
+        Positive and negative singularities::
+
+            sage: def permutations_odd_cycles(z):
+            ....:     return sqrt((1+z) / (1-z))
+            sage: ex = B.coefficients_of_generating_function(
+            ....:     permutations_odd_cycles, (1, -1,), precision=2,
+            ....: ); ex
+            sqrt(2)/sqrt(pi)*n^(-1/2) - 1/2*sqrt(1/2)/sqrt(pi)*n^(-3/2)*(-1)^n
+            + O(n^(-5/2))
+
         .. WARNING::
 
             Once singular expansions around points other than infinity