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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$, and are the answers analogous to the discrete case? Is there any combinatorial significance we can give to these integrals? Has this already been tried? 2 deleted 145 characters in body; edited title # GeneralizationofbinomialcoefficientidentitiestogammaGamma function versionsofcombinatorialidentites? I was playing with the idea of a "continuous", or "filled-in" version of Pascal's Triangle, i.e. extending the binomial coefficient$\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? 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?