Complex Binomial Coefficient C(n, m), Generalized


For C(x, y) = Bino(x, y) with x.i == 0 and y.i == 0, use the real Bino(x.r, y.r) function.

See: Real Binomial Coefficient C(n, m), Generalized

For complex x and/or y:

If y == 0 or x == y, Bino(x, 0) = Bino(x, x) = 1.

Else if y == 1, Bino(x, 1) = x.

Else if y = x − 1, Bino(x, x−1) = x.

Else if y is a real integer < 0, Bino(x, y) = 0.

Else if x−y is a real integer < 0, Bino(x, y) = 0.

Else if x is a real integer < 0, Bino(x, y) is undefined (infinite).

Else if y is a small real integer, Bino(x, y) = x*(x−1)/2*(x−2)/3* ... *(x−y+1)/y.

Else Bino(x, y) = Gam(x+1) / (Gam(y+1) * Gam(x−y+1)).

See: Binomial Coefficient -- From MathWorld
And: Wolfram Function Evaluation -- Binomial

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