### Pascal’s rule for Beta functions

27Oct13

Recently I became acquainted something called the Beta function. It is defined in terms of the Gamma function (a continuous generalization of the factorial function):

$\displaystyle \mathsf{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

The first thing I’m tempted to do here is try to write this in terms of a binomial coefficient. If we replace each $\Gamma(x)$ by the (equivalent for integers) $(x-1)!$, then we get

$\displaystyle \mathsf{B}(x, y) = \frac{(x-1)!(y-1)!}{(x+y-1)!}$

$\displaystyle = \binom{x+y-2}{x-1}^{-1}(x+y-1)^{-1}$

Now I’m more happy since I see an expression in terms of things I understand. But I would also like to know: is there a simple recurrence relation between the Beta function values, similar to Pascal’s rule $\tbinom{n+1}{k+1} = \tbinom{n}{k+1}+\tbinom{n}{k}$? It seems a little hopeless, since these guys are not only reciprocals, but they have the additional term $(x+y-1)$ complicating things.

Surprisingly, these two complications work with each other in some way and permit a really beautiful recurrence relation

$\displaystyle\mathsf{B}(x,y)=\mathsf{B}(x,y+1)+\mathsf{B}(x+1,y)$

which is even symmetric! You can prove this using the same simple algebraic approach that you can use to prove Pascal’s rule:

$\displaystyle\mathsf{B}(x,y+1)+\mathsf{B}(x+1,y)$

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

$\displaystyle = \frac{y}{x+y}\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}+\frac{x}{x+y}\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

which is just the definition of $\mathsf{B}(x, y)$ when you group the terms.

When you learn about Pascal’s rule in a combinatorics class, you learn not only the simple algebraic proof in the above vein, but also the combinatorial “proof from the book:” the number of ways to pick k+1 things from n+1 involves the initial choice of including the initial item (leaving k things to pick from n things) or excluding the initial item (leaving k+1 things to pick from n things). Is there a similarly slick proof of this Beta function identity?