Beta Function of x with y+m+1
Jump to navigation
Jump to search
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
\(\ds \map \Beta {x, y}\) | \(=\) | \(\ds \dfrac {x + y} y \map \Beta {x, y + 1}\) | Beta Function of x with y+1 by x+y over y | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {x + y}^{\overline {m + 1} } } {y^{\overline {m + 1} } } \map \Beta {x, y + m + 1}\) |
Also:
\(\ds \map \Beta {y, m + 1}\) | \(=\) | \(\ds \map \Beta {y, m} \dfrac m {y + m}\) | Beta Function of x with y+1 by x+y over y | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {m!} {\paren {y + 1}^{\overline m} } \map \Beta {y, 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {m!} {y^{\overline {m + 1} } }\) |
and:
\(\ds \map \Beta {x + y, m + 1}\) | \(=\) | \(\ds \map \Beta {x + y, m} \dfrac m {x + y + m}\) | Beta Function of x with y+1 by x+y over y | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {m!} {\paren {x + y}^{\overline {m + 1} } }\) |
Hence:
\(\ds \map \Beta {x, y}\) | \(=\) | \(\ds \dfrac {\map \Beta {y, m + 1} } {\map \Beta {x + y, m + 1} } \map \Beta {x, y + m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map {\Gamma_m} y} {m^y} \dfrac {m^{x + y} } {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $41 \ \text{(a)}$