We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient identities have generalizations to this setting? Just as two simple examples, we have


$\sum\_{k=0}^n \binom{n}{k} = 2^n$ and $\sum\_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}$


What are $\int\_0^x \binom{x}{y} dy$ and $\int\_0^x \binom{x}{y}^2 dy$, and are the answers analogous to the discrete case? Is there any combinatorial significance we can give to these integrals? Has this already been tried?



Gamma function versions of combinatorial identites?
Chapter 5.5 of *Concrete Mathematics* discusses generalizing binomial coefficient identities to the Gamma function. It doesn't discuss the two integrals you mention, though.


Doing a bit of thinking on my own, if $n$ is a positive integer then
$$\int\_{z=0}^n \binom{n}{z} dz = \int\_{z=0}^n \frac{n! dz}{\Gamma(1+z) \Gamma(n+1-z)}$$
$$\int\_{z=0}^{n} \frac{n! dz}{(n-z)(n-1-z) \cdots (1-z) \Gamma(1-z) \Gamma(1+z)}.$$


We have $\Gamma(1+z) \Gamma(1-z) = \pi z/\sin (\pi z)$, if I haven't made any dumb errors, so this is
$$\int\_{0}^n \frac{ n! \sin (\pi z) \ dz}{\pi z (n-z)(n-1-z) \cdots (1-z)}.$$


I suspect this integrand does not have an elementary anti-derivative, because it reminds me of $\int \sin t \ dt/t$. But there might be some special trick which would let you compute the integral between these specific bounds.


