A binomial at a pole of its denominator is zero

BREAKING-CHANGES.md

@@ -623,13 +623,15 @@ It is the forward difference with a step of 1, as Mathematica's `DifferenceDelta
 
 **Not silent: what threw is a number.** `binomial(n, k)` with a whole `n` and a `k` that is not whole was left as
 written, so `EvalNumerical` threw `CannotEvalException`. It is the gamma ratio `Γ(n + 1)/(Γ(k + 1) Γ(n - k + 1))`, as
-for any other `n`, and for a negative whole `n` it has no value: `Γ(n + 1)` has a pole there
-([#1411](https://github.com/asc-community/AngouriMath/issues/1411)).
+for any other `n`, and for a negative whole `n` it has no value: `Γ(n + 1)` has a pole there. Where `n - k` is a
+negative whole number, a pole of `Γ(n - k + 1)` in the denominator, it is `0`, the reciprocal gamma vanishing there;
+it was left as written ([#1411](https://github.com/asc-community/AngouriMath/issues/1411)).
 
 | Input | Was (2.6.0) | Now |
 |---|---|---|
 | `"binomial(3, 1/2)".ToEntity().EvalNumerical()` | `CannotEvalException` | `2.0371832715762602978…`, `32/(5 pi)` |
 | `"binomial(-3, 1/2)".ToEntity().EvalNumerical()` | `CannotEvalException` | `NaN` |
+| `"binomial(1/2, 3/2)".ToEntity().Simplify()` | `binomial(1/2, 3/2)` | `0` |
 
 ### The harmonic numbers
 

Sources/AngouriMath/Functions/Evaluation/Evaluation.Continuous/Evaluation.Continuous.Arithmetics.Classes.cs

@@ -568,6 +568,9 @@ protected override Entity InnerSimplify(bool isExact) =>
                         (_, Integer { IsZero: true }) => Integer.Create(1),
                         // A k that is not whole has no value with a negative whole n, a pole of Γ(n + 1).
                         (Integer { EInteger.Sign: < 0 }, Complex bottom) when bottom is not Integer => MathS.NaN,
+                        // And is 0 where n - k is a negative whole number, a pole of Γ(n - k + 1), where the
+                        // reciprocal gamma is 0: binomial(1/2, 3/2) is 0.
+                        (Number top, Number bottom) when bottom is not Integer && (top - bottom).Evaled is Integer { EInteger.Sign: < 0 } => Integer.Zero,
                         // Through the gamma function only numerically: there is no exact form. A whole n
                         // takes it too: binomial(3, 1/2) is 32/(5 pi).
                         (Complex top, Complex bottom) when !isExact

Sources/Tests/UnitTests/Convenience/BinomialTest.cs

@@ -83,6 +83,22 @@ public void AWholeUpperArgumentWithAFractionalLowerOneEvaluatesThroughTheGammaFu
             Assert.True(value.EDecimal.Subtract(PeterO.Numbers.EDecimal.FromString(expected)).Abs().CompareTo(PeterO.Numbers.EDecimal.FromString("1e-25")) < 0, $"{input} evaluated to {value}");
         }
 
+        /// <summary>
+        /// Where <c>n - k</c> is a negative whole number, <c>Γ(n - k + 1)</c> in the denominator has a pole
+        /// and the binomial is 0, as mpmath has it: the reciprocal gamma vanishes there. It was left
+        /// unevaluated.
+        /// </summary>
+        [Theory]
+        [InlineData("binomial(1/2, 3/2)")]
+        [InlineData("binomial(5/2, 7/2)")]
+        [InlineData("binomial(1/2, 5/2)")]
+        [InlineData("binomial(3/2 + i, 5/2 + i)")]
+        public void AtAPoleOfTheDenominatorItIsZero(string input)
+        {
+            Assert.Equal(Entity.Number.Integer.Zero, input.ToEntity().InnerSimplified);
+            Assert.Equal(Entity.Number.Integer.Zero, input.ToEntity().EvalNumerical());
+        }
+
         [Fact]
         public void ANegativeWholeUpperArgumentWithAFractionalLowerOneHasNoValue()
             => Assert.Equal(MathS.NaN, "binomial(-3, 1/2)".ToEntity().EvalNumerical());