Define, for $\displaystyle x \geq y \geq 0$

$\displaystyle \Lambda(x,y)=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma( x-y+1)}$

Then proof for all $\displaystyle n>0$

$\displaystyle \int^n_{0}\Lambda(n,x)dx=2^n$

And also prove for $\displaystyle 0<p<1$

$\displaystyle \int^ \infty_{-\infty}p^x(1-p)^{n-x}\Lambda(n,x)dx=1$

I once asked someone this question and they said it looks like the Beta Function, thus maybe that will help.

If you are curious how I came to such a conjecture is because I was trying to describe a mathematical formula for the Normal Distribution and these two:

$\displaystyle \Lambda(n,x)$

and

$\displaystyle p^x(1-p)^{n-x}\Lambda(n,x)$ where $\displaystyle p$ is the probability.

Graph curves which look like the Normal Distribution, furthermore the second one is a density curve this is why its Area is one.