binomial(x,k) for non-integer x (#48124)

* binomial(x,k) for non-integer x

* correct PR number in NEWS

* return 0 for k < 0

* comment on further generalizations

NEWS.md

@@ -30,8 +30,9 @@ New library functions
 
 New library features
 --------------------
-The `initialized=true` keyword assignment for `sortperm!` and `partialsortperm!`
-is now a no-op ([#47979]). It previously exposed unsafe behavior ([#47977]).
+* The `initialized=true` keyword assignment for `sortperm!` and `partialsortperm!`
+  is now a no-op ([#47979]). It previously exposed unsafe behavior ([#47977]).
+* `binomial(x, k)` now supports non-integer `x` ([#48124]).
 
 Standard library changes
 ------------------------

base/intfuncs.jl

@@ -1102,3 +1102,34 @@ Base.@assume_effects :terminates_locally function binomial(n::T, k::T) where T<:
     end
     copysign(x, sgn)
 end
+
+"""
+    binomial(x::Number, k::Integer)
+
+The generalized binomial coefficient, defined for `k ≥ 0` by
+the polynomial
+```math
+\\frac{1}{k!} \\prod_{j=0}^{k-1} (x - j)
+```
+When `k < 0` it returns zero.
+
+For the case of integer `x`, this is equivalent to the ordinary
+integer binomial coefficient
+```math
+\\binom{n}{k} = \\frac{n!}{k! (n-k)!}
+```
+
+Further generalizations to non-integer `k` are mathematically possible, but
+involve the Gamma function and/or the beta function, which are
+not provided by the Julia standard library but are available
+in external packages such as [SpecialFunctions.jl](https://github.com/JuliaMath/SpecialFunctions.jl).
+
+# External links
+* [Binomial coefficient](https://en.wikipedia.org/wiki/Binomial_coefficient) on Wikipedia.
+"""
+function binomial(x::Number, k::Integer)
+    k < 0 && return zero(x)/one(k)
+    # we don't use prod(i -> (x-i+1), 1:k) / factorial(k),
+    # and instead divide each term by i, to avoid spurious overflow.
+    return prod(i -> (x-(i-1))/i, OneTo(k), init=oneunit(x)/one(k))
+end

test/intfuncs.jl

@@ -554,6 +554,14 @@ end
     for x in ((false,false), (false,true), (true,false), (true,true))
         @test binomial(x...) == (x != (false,true))
     end
+
+    # binomial(x,k) for non-integer x
+    @test @inferred(binomial(10.0,3)) === 120.0
+    @test @inferred(binomial(10//1,3)) === 120//1
+    @test binomial(2.5,3) ≈ 5//16 === binomial(5//2,3)
+    @test binomial(2.5,0) == 1.0
+    @test binomial(35.0, 30) ≈ binomial(35, 30) # naive method overflows
+    @test binomial(2.5,-1) == 0.0
 end
 
 # concrete-foldability