Beta Function of x with y+m+1

Theorem
Let $\Beta \left({x, y}\right)$ denote the Beta function.

Then:
 * $\Beta \left({x, y}\right) = \dfrac {\Gamma_m \left({y}\right) m^x} {\Gamma_m \left({x + y}\right)} \Beta \left({x, y + m + 1}\right)$

where $\Gamma_m$ is the partial Gamma function:
 * $\displaystyle \Gamma_m \left({y}\right) := \frac {m^y m!} {y \left({y + 1}\right) \left({y + 2}\right) \cdots \left({y + m}\right)}$

Proof
From Partial Gamma Function expressed as Integral:

Hence: