Trac #22004: Allow algorithm='sympy' in symbolic_sum function

I want this to work:
{{{
sage: n = var('n')
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='sympy')
}}}

See [https://ask.sagemath.org/question/35839/sage-incorrectly-evaluates-
series/ this question on ask.sagemath.org]

URL: https://trac.sagemath.org/22004
Reported by: slabbe
Ticket author(s): Sébastien Labbé
Reviewer(s): Ralf Stephan

Author: Release Manager

src/sage/calculus/calculus.py

@@ -451,6 +451,8 @@ def symbolic_sum(expression, v, a, b, algorithm='maxima', hold=False):
 
       - ``'giac'`` - (optional) use Giac
 
+      - ``'sympy'`` - use SymPy
+
     - ``hold`` - (default: ``False``) if ``True`` don't evaluate
 
     EXAMPLES::
@@ -554,6 +556,22 @@ def symbolic_sum(expression, v, a, b, algorithm='maxima', hold=False):
         sage: symbolic_sum(1/(1+k^2), k, -oo, oo, algorithm = 'giac')           # optional - giac
         (pi*e^(2*pi) - pi*e^(-2*pi))/(e^(2*pi) + e^(-2*pi) - 2)
 
+    SymPy can't solve that summation::
+
+        sage: symbolic_sum(1/(1+k^2), k, -oo, oo, algorithm = 'sympy')
+        Traceback (most recent call last):
+        ...
+        AttributeError: Unable to convert SymPy result (=Sum(1/(k**2 + 1),
+        (k, -oo, oo))) into Sage
+
+    But SymPy can do this one for which Maxima is broken (see
+    :trac:`22005`)::
+
+        sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='sympy')
+        1/64*pi^2
+        sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity)
+        1/64*pi^2 - 1/12
+
     Use Maple as a backend for summation::
 
         sage: symbolic_sum(binomial(n,k)*x^k, k, 0, n, algorithm = 'maple')      # optional - maple
@@ -632,6 +650,16 @@ def symbolic_sum(expression, v, a, b, algorithm='maxima', hold=False):
             raise ValueError("Giac cannot make sense of: %s" % sum)
         return result.sage()
 
+    elif algorithm == 'sympy':
+        expression,v,a,b = [expr._sympy_() for expr in (expression, v, a, b)]
+        from sympy import summation
+        result = summation(expression, (v, a, b))
+        try:
+            return result._sage_()
+        except AttributeError:
+            raise AttributeError("Unable to convert SymPy result (={}) into"
+                    " Sage".format(result))
+
     else:
         raise ValueError("unknown algorithm: %s" % algorithm)
 

src/sage/misc/functional.py

@@ -420,6 +420,8 @@ def symbolic_sum(expression, *args, **kwds):
 
       - ``'giac'`` - (optional) use Giac
 
+      - ``'sympy'`` - use SymPy
+
     EXAMPLES::
 
         sage: k, n = var('k,n')

src/sage/symbolic/expression.pyx

@@ -11452,6 +11452,7 @@ cdef class Expression(CommutativeRingElement):
 
                 - ``'giac'`` - (optional) use Giac
 
+                - ``'sympy'`` - use SymPy
 
         EXAMPLES::