Pascal’s rule for Beta functions
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):
The first thing I’m tempted to do here is try to write this in terms of a binomial coefficient. If we replace each by the (equivalent for integers) , then we get
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 ? It seems a little hopeless, since these guys are not only reciprocals, but they have the additional term complicating things.
Surprisingly, these two complications work with each other in some way and permit a really beautiful recurrence relation
which is even symmetric! You can prove this using the same simple algebraic approach that you can use to prove Pascal’s rule:
which is just the definition of 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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