In the spirit of the previous post, one thing that you do in calculus of variations – a lot – is integration by parts. It’s incredible how useful that one trick is. See, they teach you these things in calculus for a reason!
But, I was fiddling around yesterday, and got this nice result, applying many integrations by parts. For integers n, m,
Which I thought was kind of nice.
That’s the Beta function. And if you write the factorials as gamma functions, your equation holds for real numbers whenever n and m are such that the integral converges (that is, they’re greater than -1).
Comment by Michael Lugo — July 24, 2009 @ 8:25 pm
Instead of repeated integration by parts you can integrate (x y + 1-x) ^ (m+n) dx from 0 to 1 (y is a free parameter), expand both the integrand and the computed integral in powers of y and equate coefficients.
This of course is just computing all the integrals with a fixed m+n simultaneously by computing their generating function, usually a good trick.
Comment by Omar — July 26, 2009 @ 4:17 am
I posted before but it didn’t publish…
Anyways, I repeatedly integrated x^n * dx term and differentiated the (1-x)^m term. So I get m(m-1)(m-2)… for numerator and (n+1)(n+2)(n+3)… for denominator. Simplifying this, I get the same numerator as yours, but my denominator is (n + infinity)! <–factorial, not exclamation. How does this infinity translate to m+1?
Comment by Kitten — July 28, 2009 @ 7:40 am
Nevermind I figured it out.
Comment by Kitten — July 28, 2009 @ 2:44 pm
I’ve seen a generalization of this, I think it goes thusly: the integration of the monomial x1^p1 … xn^pn over the simplex is p1!…pn!/(p1+…+pn)!
Comment by Alex — August 18, 2009 @ 10:11 pm