Beta Function of x+1 with y plus Beta Function of x with y+1
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Theorem
Let $\map \Beta {x, y}$ denote the Beta function.
Then:
- $\map \Beta {x + 1, y} + \map \Beta {x, y + 1} = \map \Beta {x, y}$
Proof
By definition:
\(\ds \map \Beta {x + 1, y} + \map \Beta {x, y + 1}\) | \(=\) | \(\ds \int_0^1 t^x \left({1 - t}\right)^{y - 1} \rd t + \int_0^1 t^{x - 1} \left({1 - t}\right)^y \rd t\) | Definition of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \left({t^x \left({1 - t}\right)^{y - 1} + t^{x - 1} \left({1 - t}\right)^y}\right) \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 t^{x - 1} \left({1 - t}\right)^{y - 1} \left({t + 1 - t}\right) \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 t^{x - 1} \left({1 - t}\right)^{y - 1} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Beta {x, y}\) |
$\blacksquare$
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 $40 \ \text{(b)}$