Binomial Coefficient of Real Number with Half
Jump to navigation
Jump to search
Theorem
- $\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$
where $\dbinom r {1 / 2}$ denotes a binomial coefficient.
Proof
\(\ds \dbinom r {1 / 2}\) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {\map \Gamma {\rho + 1} } {\map \Gamma {\frac 1 2 + 1} \map \Gamma {\rho - \frac 1 2 + 1} }\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {\map \Gamma {\rho + 1} } {\frac 1 2 \map \Gamma {\frac 1 2} \map \Gamma {\rho + \frac 1 2} }\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2 \map \Gamma {\rho + 1} } {\sqrt \pi \, \map \Gamma {\rho + \frac 1 2} }\) | Gamma Function of One Half |
Legendre's Duplication Formula gives:
- $\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$
and so:
- $(1): \quad \map \Gamma {\rho + \dfrac 1 2} = \dfrac {2^{1 - 2 \rho} \sqrt \pi \, \map \Gamma {2 \rho} } {\map \Gamma \rho}$
Hence:
\(\ds \dbinom r {1 / 2}\) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2 \map \Gamma {\rho + 1} } {\sqrt \pi \, \dfrac {2^{1 - 2 \rho} \sqrt \pi \, \map \Gamma {2 \rho} } {\map \Gamma \rho} }\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2 \map \Gamma {\rho + 1} \map \Gamma \rho} {\pi 2^{1 - 2 \rho} \map \Gamma {2 \rho} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2^{2 \rho - 1} 2 \map \Gamma {\rho + 1} \map \Gamma \rho} {\pi \map \Gamma {2 \rho} }\) | rearranging and simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2^{2 \rho} } {\pi \dfrac {\map \Gamma {2 \rho} } {\map \Gamma \rho \map \Gamma {\rho + 1} } }\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2^{2 \rho} } {\pi \dfrac {\rho \map \Gamma {2 \rho + 1} } {2 \rho \map \Gamma {\rho + 1} \map \Gamma {\rho + 1} } }\) | Gamma Difference Equation twice | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\rho \mathop \to r} \dfrac {2^{2 \rho + 1} } {\pi \dfrac {\map \Gamma {2 \rho + 1} } {\map \Gamma {\rho + 1} \map \Gamma {\rho + 1} } }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}\) | Definition of Binomial Coefficient |
$\blacksquare$
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 $44$