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Mathematica returns values for non-integer arguments passed to Binomial. What is the definition of Binomial for such continuous arguments?


Binomial[9, 2]
Binomial[9, 2.3212312]
Binomial[9, 3]
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The factorials are just replaced with gamma functions for noninteger arguments, since $n!=\Gamma(n+1)$. – J. M. May 17 '13 at 11:31
Your question is not clear: are you asking what is the definition of Binomial for non-integer arguments? Or are you asking how this value is computed? The definition is in the documentation under details. I don't know how it's computed, but it's probably not directly using the formula in terms of $\Gamma$ functions as that would result in the ratio of very large numbers. – Szabolcs May 17 '13 at 14:22
I agree @Szabolcs, my question wasn't very clear. I'm satisfied by knowing that the Gamma function is used for non-integer input as pointed out by both J.M. and as stated in the documentation -- I honestly didn't read the documentation carefully enough. – Name May 17 '13 at 15:08
@Name I made the question unambiguous. Please review the edit. – Szabolcs May 17 '13 at 15:14
@Szabolcs, "'s probably not directly using the formula in terms of $\Gamma$ functions...", yes, likely combinations of LogGamma[] are used before exponentiating. – J. M. May 17 '13 at 15:28
up vote 7 down vote accepted

Mathematica defines Binomial for non-integer inputs as follows:

$$ \binom{n}{m} = \frac{\Gamma(n+1)}{\Gamma(m+1)\Gamma(n-m+1)} $$

You'll find this under the Details section on the documentation page of Binomial.

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