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?

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2 Responses to “Pascal’s rule for Beta functions”

  1. 1 Brian

    It seems to me that if you wanted a “similarly slick” proof for the beta function identity then you’d need a slick definition for the beta function first. Is there one?

  2. One try: for integer values, the Beta function B(x, y) is the inverse of the multinomial coefficient for (x-1, y-1, 1). I.e. if we arrange the numbers from -x+1 to y-1 in a random order, it is the probability that the negatives come before 0 and the positives come after 0. Could this help?

    For non-integer values, I wouldn’t know where to start, but doing the integer case would be awesome enough.


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