[concrete/summations] Implement some binomial sums.

Specifically, in the evaluation method using hypergeometric functions,
when encountering a finite sum from a to b, try to evaluate it by summing
from a to infinity and subtracting the sum from b+1 to infinity.

Fix a problem where _eval_sum_hyper substitutes zero in the expression
and, if it gets zero, adds one and starts over. This yields to an
inifinite loop if the expression is identically zero under the correct
assumptions.

Extend binomial.eval() to recognise that binomial(n, k) is always zero
if k is bigger than n.

Add tests.

sympy/concrete/summations.py

@@ -479,13 +479,15 @@ def eval_sum_symbolic(f, limits):
 def _eval_sum_hyper(f, i, a):
     """ Returns (res, cond). Sums from a to oo. """
     from sympy.functions import hyper
-    from sympy.simplify import hyperexpand, hypersimp, fraction
+    from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
     from sympy.polys.polytools import Poly, factor
 
     if a != 0:
         return _eval_sum_hyper(f.subs(i, i + a), i, 0)
 
     if f.subs(i, 0) == 0:
+        if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
+            return S(0), True
         return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
 
     hs = hypersimp(f, i)
@@ -532,7 +534,15 @@ def eval_sum_hyper(f, (i, a, b)):
             if res is not None:
                 return Piecewise(res, (Sum(f, (i, a, b)), True))
         else:
-            return None
+            res1 = _eval_sum_hyper(f, i, a)
+            res2 = _eval_sum_hyper(f, i, b + 1)
+            if res1 is None or res2 is None:
+                return None
+            (res1, cond1), (res2, cond2) = res1, res2
+            cond = And(cond1, cond2)
+            if cond is False:
+                return None
+        return Piecewise((res1 - res2, cond), (Sum(f, (i, a, b)), True))
 
     if a == -oo:
         res1 = _eval_sum_hyper(f.subs(i, -i), i, 1)

sympy/concrete/tests/test_sums_products.py

@@ -289,6 +289,9 @@ def test_hypersum():
     assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
     assert s.args[0].args[1] == (abs(1/x) < 1)
 
+    m = Symbol('n', integer=True, positive=True)
+    assert summation(binomial(m, k), (k, 0, m)) == 2**m
+
 def test_issue_1071():
     assert summation(1/factorial(k), (k, 0, oo)) == E
 

sympy/functions/combinatorial/factorials.py

@@ -463,6 +463,8 @@ def eval(cls, n, k):
                     return result
         elif k.is_negative:
             return S.Zero
+        elif (n - k).simplify().is_negative:
+            return S.Zero
         else:
             d = n - k
 

sympy/functions/combinatorial/tests/test_comb_factorials.py

@@ -120,6 +120,7 @@ def test_binomial():
     assert binomial(n, u) == 0
     assert binomial(n, v).func == binomial
     assert binomial(n, k).func == binomial
+    assert binomial(n, n + v) == 0
 
 def test_binomial_diff():
     n = Symbol('n', integer=True)