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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    $\begingroup$ The factorials are just replaced with gamma functions for noninteger arguments, since $n!=\Gamma(n+1)$. $\endgroup$ – J. M.'s ennui May 17 '13 at 11:31
  • $\begingroup$ 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. $\endgroup$ – Szabolcs May 17 '13 at 14:22
  • $\begingroup$ 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. $\endgroup$ – Name May 17 '13 at 15:08
  • $\begingroup$ @Name I made the question unambiguous. Please review the edit. $\endgroup$ – Szabolcs May 17 '13 at 15:14
  • $\begingroup$ @Szabolcs, "...it's probably not directly using the formula in terms of $\Gamma$ functions...", yes, likely combinations of LogGamma[] are used before exponentiating. $\endgroup$ – J. M.'s ennui May 17 '13 at 15:28

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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