# 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

 $\displaystyle \map \Beta {x, y}$ $=$ $\displaystyle \dfrac {x + y} y \map \Beta {x, y + 1}$ Beta Function of x with y+1 by x+y over y $\displaystyle$ $=$ $\displaystyle \dfrac {\paren {x + y}^{\overline {m + 1} } } {y^{\overline {m + 1} } } \map \Beta {x, y + m + 1}$

Also:

 $\displaystyle \map \Beta {y, m + 1}$ $=$ $\displaystyle \map \Beta {y, m} \dfrac m {y + m}$ Beta Function of x with y+1 by x+y over y $\displaystyle$ $=$ $\displaystyle \dfrac {m!} {\paren {y + 1}^{\overline m} } \map \Beta {y, 1}$ $\displaystyle$ $=$ $\displaystyle \dfrac {m!} {y^{\overline {m + 1} } }$

and:

 $\displaystyle \map \Beta {x + y, m + 1}$ $=$ $\displaystyle \map \Beta {x + y, m} \dfrac m {x + y + m}$ Beta Function of x with y+1 by x+y over y $\displaystyle$ $=$ $\displaystyle \dfrac {m!} {\paren {x + y}^{\overline {m + 1} } }$

Hence:

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

$\blacksquare$