Merge pull request #1261 from ness01/summation

Summation improvements.

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)

sympy/simplify/hyperexpand.py

@@ -1932,7 +1932,7 @@ def _hyperexpand(ip, z, ops0=[], z0=Dummy('z0'), premult=1, prem=0,
     is multiplied by premult. Then ops0 is applied.
     premult must be a*z**prem for some a independent of z.
     """
-    from sympy.simplify import powdenest, simplify, polarify
+    from sympy.simplify import powdenest, simplify, polarify, unpolarify
     z = polarify(z, subs=False)
     if rewrite == 'default':
         rewrite = 'nonrepsmall'
@@ -1977,7 +1977,7 @@ def carryout_plan(f, ops):
         debug('  Recognised polynomial.')
         p = apply_operators(res, ops, lambda f: z0*f.diff(z0))
         p = apply_operators(p*premult, ops0, lambda f: z0*f.diff(z0))
-        return simplify(p).subs(z0, z)
+        return unpolarify(simplify(p).subs(z0, z))
 
     # Try to recognise a shifted sum.
     p = S(0)
@@ -1993,7 +1993,6 @@ def carryout_plan(f, ops):
     p = simplify(p).subs(z0, z)
 
     # Try special expansions early.
-    from sympy import unpolarify
     if unpolarify(z) in [1, -1] and (len(nip.ap), len(nip.bq)) == (2, 1):
         f = build_hypergeometric_formula(nip)
         r = carryout_plan(f, ops).replace(hyper, hyperexpand_special)

sympy/simplify/tests/test_hyperexpand.py

@@ -934,3 +934,7 @@ def test_prudnikov_fail_other():
 
     # XXX this does not *evaluate* right??
     assert can_do([], [a, a + S.Half, 2*a-1])
+
+def test_bug():
+    h = hyper([-1, 1], [z], -1)
+    assert hyperexpand(h) == (z + 1)/z