Beta Function of x with y+m+1

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

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

where $\Gamma_m$ is the partial Gamma function:
 * $\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \cdots \paren {y + m} }$

Proof
Also:

and:

Hence: