Beta Function of x with y+m+1/Proof 2
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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:
| \(\ds \map {\Gamma_m} y\) | \(=\) | \(\ds \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \cdots \paren {y + m} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {m^y m!} { y^{\overline {m + 1} } }\) |
Proof
| \(\ds \map \Beta {x, y}\) | \(=\) | \(\ds \dfrac {x + y} y \map \Beta {x, y + 1}\) | Beta Function of x with y+1 by x+y over y | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\dfrac {x + y} y } \paren {\dfrac {x + y + 1} {y + 1} } \map \Beta {x, y + 1 + 1}\) | applying Beta Function of x with y+1 by x+y over y a second time | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {x + y}^{\overline {m + 1} } } {y^{\overline {m + 1} } } \map \Beta {x, y + m + 1}\) | after $m + 1$ times: rising factorial: $y^{\overline {m + 1} } = y \paren {y + 1} \cdots \paren {y + m}$ |
Hence:
| \(\ds \map \Beta {x, y}\) | \(=\) | \(\ds \dfrac {\paren {x + y}^{\overline {m + 1} } } {y^{\overline {m + 1} } } \map \Beta {x, y + m + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\dfrac {m^x m^y m!} {m^x m^y m!} } \dfrac {\paren {\dfrac 1 {y^{\overline {m + 1} } } } } {\paren {\dfrac 1 {\paren {x + y}^{\overline {m + 1} } } } } \map \Beta {x, y + m + 1}\) | multiplying by $1$ and rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {m^x m^y m!} {y^{\overline {m + 1} } } } } {\paren {\dfrac {m^x m^y m!} {\paren {x + y}^{\overline {m + 1} } } } } \map \Beta {x, y + m + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {m^y m!} {y^{\overline {m + 1} } } } m^x } {\paren {\dfrac {m^x m^y m!} {\paren {x + y}^{\overline {m + 1} } } } } \map \Beta {x, y + m + 1}\) | isolating $m^x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}\) | $\map {\Gamma_m} y = \dfrac {m^y m!} { y^{\overline {m + 1} } }$ and $\map {\Gamma_m} {x + y} = \dfrac {m^{x + y} m!} { \paren {x + y}^{\overline {m + 1} } }$ |
$\blacksquare$