Binomial Coefficient of Real Number with Half

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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

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