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
| \(\displaystyle \Beta \left({x, y}\right)\) | \(=\) | \(\displaystyle \dfrac {x + y} y \Beta \left({x, y + 1}\right)\) | $\quad$ Beta Function of x with y+1 by x+y over y | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\left({x + y}\right)^{\overline {m + 1} } } {y^{\overline {m + 1} } } \Beta \left({x, y + m + 1}\right)\) | $\quad$ | $\quad$ |
Also:
| \(\displaystyle \Beta \left({y, m + 1}\right)\) | \(=\) | \(\displaystyle \Beta \left({y, m}\right) \dfrac m {y + m}\) | $\quad$ Beta Function of x with y+1 by x+y over y | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {m!} {\left({y + 1}\right)^{\overline m} } \Beta \left({y, 1}\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {m!} {y^{\overline {m + 1} } }\) | $\quad$ | $\quad$ |
and:
| \(\displaystyle \Beta \left({x + y, m + 1}\right)\) | \(=\) | \(\displaystyle \Beta \left({x + y, m}\right) \dfrac m {x + y + m}\) | $\quad$ Beta Function of x with y+1 by x+y over y | $\quad$
}} | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {m!} {\left({x + y}\right)^{\overline {m + 1} } }\) | $\quad$ | $\quad$ |
Hence:
| \(\displaystyle \Beta \left({x, y}\right)\) | \(=\) | \(\displaystyle \dfrac {\Beta \left({y, m + 1}\right)} {\Beta \left({x + y, m + 1}\right)} \Beta \left({x, y + m + 1}\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\Gamma_m \left({y}\right)} {m^y} \dfrac {m^{x + y} } {\Gamma_m \left({x + y}\right)} \Beta \left({x, y + m + 1}\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\Gamma_m \left({y}\right) m^x} {\Gamma_m \left({x + y}\right)} \Beta \left({x, y + m + 1}\right)\) | $\quad$ | $\quad$ |
$\blacksquare$
In particular: Justification for certain of the steps above.
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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 $41 \ \text{(a)}$
