Binomial Coefficient of Half/Corollary

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Theorem

Let $k \in \Z_{\ge 0}$.

$\dbinom {\frac 1 2} k = \dfrac {\paren {-1}^{k - 1} } {2^{2 k - 1} \paren {2 k - 1} } \dbinom {2 k - 1} k - \delta_{k 0}$

where:

$\dbinom {\frac 1 2} k$ denotes a binomial coefficient
$\delta_{k 0}$ denotes the Kronecker delta.


Proof

When $k > 0$ we have:

\(\ds \dfrac {\paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dbinom {2 k} k\) \(=\) \(\ds \dfrac {\paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dfrac {2 k} {2 k - k} \dbinom {2 k - 1} k\) Factors of Binomial Coefficient: Corollary 1
\(\ds \) \(=\) \(\ds \dfrac {\paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dfrac {2 k} k \dbinom {2 k - 1} k\)
\(\ds \) \(=\) \(\ds \dfrac {2 \paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dbinom {2 k - 1} k\)


When $k = 0$ we have:

\(\ds \dfrac {\paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dbinom {2 k} k\) \(=\) \(\ds \dfrac {\paren {-1}^{-1} } {-1} \dbinom 0 0\)
\(\ds \) \(=\) \(\ds 1\)


while:

\(\ds \dfrac {\paren {-1}^{k - 1} } {2^{2 k - 1} \paren {2 k - 1} } \dbinom {2 k - 1} k\) \(=\) \(\ds \dfrac {\paren {-1}^{-1} } {2^{-1} \paren {-1} } \dbinom {-1} 0\)
\(\ds \) \(=\) \(\ds \dfrac 1 {2^{-1} } \dbinom {-1} 0\)
\(\ds \) \(=\) \(\ds 2 \dbinom {-1} 0\)
\(\ds \) \(=\) \(\ds 2 \paren {-1}^0 \dbinom {0 - \paren {-1} - 1} 0\) Negated Upper Index of Binomial Coefficient
\(\ds \) \(=\) \(\ds 2 \dbinom 0 0\)
\(\ds \) \(=\) \(\ds 2\)


Hence:

$(1): \quad \dbinom {\frac 1 2} k = \dfrac {2 \paren {-1}^{k - 1} } {4^k \paren {2 k - 1} } \dbinom {2 k - 1} k - \delta_{k 0}$

$\blacksquare$


Sources

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