This question is more precisely about evaluation with a computer, of a binomial coefficient of the form $ \binom{x}{m}$ where $x$ is a real number and $m$ a rational integer.

The reason why I ask is that I found out recently that sage is using the naive definition with the $\Gamma$ function, which means that it gets as a result NaN (not-a-number) with quite small parameters, for which the real result is pretty reasonable and should have been given (see the bug report).

I have proposed to change the implementation by returning zero in more cases than it already does, to reduce to a situation $\binom{x}{m}$ with $x\geq m\geq 0$, so we can write $x=m+k+u$ with $k$ a natural integer and $u\in[0;1[$, then computing the quotient $\Gamma(x+1)/\Gamma(m+1)$ with a Pochhammer symbol times the quotient $\Gamma(m+1+u)/\Gamma(m+1)$. For that last quotient, I was proposing a direct computation for small $m$ and a polynomial expansion in $u$ for big $m$.

There are two problems with this approach:

- I don't really know how big the error is, which for a numerical computation is a pretty big issue ;
- I used the naive code as a starting point, and added naive ideas to the mix : there may exist better approaches (it's also because of this point that I didn't try to evaluate the error more precisely).

It would be surprising if there existed no algorithm for this kind of computations, given how important those coefficients are in various situations...