Beta Function of x+1 with y plus Beta Function of x with y+1

Theorem

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

Then:

$\map \Beta {x + 1, y} + \map \Beta {x, y + 1} = \map \Beta {x, y}$

By definition:

$\ds \map \Beta {x + 1, y} + \map \Beta {x, y + 1}$

$=$

$\ds \int_0^1 t^x \left({1 - t}\right)^{y - 1} \rd t + \int_0^1 t^{x - 1} \left({1 - t}\right)^y \rd t$

Definition of Beta Function

$\ds $

$=$

$\ds \int_0^1 \left({t^x \left({1 - t}\right)^{y - 1} + t^{x - 1} \left({1 - t}\right)^y}\right) \rd t$

$\ds $

$=$

$\ds \int_0^1 t^{x - 1} \left({1 - t}\right)^{y - 1} \left({t + 1 - t}\right) \rd t$

$\ds $

$=$

$\ds \int_0^1 t^{x - 1} \left({1 - t}\right)^{y - 1} \rd t$

$\ds $

$=$

$\ds \map \Beta {x, y}$

$\blacksquare$