Beta Function of x with y+1 by x+y over y

Theorem
Let $\map \Beta {x, y}$ denote the Beta function.

Then:
 * $\map \Beta {x, y} = \dfrac {x + y} y \map \Beta {x, y + 1}$

Proof
By definition of Beta function:


 * $\ds \map \Beta {x + 1, y} = \int_0^1 t^x \paren {1 - t}^{y - 1} \rd t$

With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Hence: