MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 3 simplified the question, removed unnecessary bits

I was playing with the idea of a "continuous", or "filled-in" version of Pascal's Triangle, i.e. extending

We can extend the binomial coefficient $\binom{n}{k}$ to (at least for my purposes) $n,k\in\mathbb{R}$, $0\leq k\leq n$, so I jumped straight to the gamma function, the obvious choice being $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$ (this is also listed on Wikipedia as how to extend the binomial coefficients to $\mathbb{R}$ or $\mathbb{C}$). Okay, straightforward enough so far - but my question is, do \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?
show/hide this revision's text 2 deleted 145 characters in body; edited title

Generalization of binomial coefficient identities to gamma Gamma function versions of combinatorial identites?

I was playing with the idea of a "continuous", or "filled-in" version of Pascal's Triangle, i.e. extending the binomial coefficient $\binom{n}{k}$ to (at least for my purposes) $n,k\in\mathbb{R}$, $0\leq k\leq n$, so I jumped straight to the gamma function, the obvious choice being $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$ (this is also listed on Wikipedia as how to extend the binomial coefficients to $\mathbb{R}$ or $\mathbb{C}$). Okay, straightforward enough so far - but my question is, 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?

Unfortunately, I don't have the faintest idea of how to compute these integrals (I really need to learn complex analysis already). Any ideas? Has this already been tried?
show/hide this revision's text 1

Generalization of binomial coefficient identities to gamma function

I was playing with the idea of a "continuous", or "filled-in" version of Pascal's Triangle, i.e. extending the binomial coefficient $\binom{n}{k}$ to (at least for my purposes) $n,k\in\mathbb{R}$, $0\leq k\leq n$, so I jumped straight to the gamma function, the obvious choice being $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$ (this is also listed on Wikipedia as how to extend the binomial coefficients to $\mathbb{R}$ or $\mathbb{C}$). Okay, straightforward enough so far - but my question is, 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?

Unfortunately, I don't have the faintest idea of how to compute these integrals (I really need to learn complex analysis already). Any ideas? Has this already been tried?