Binomial Coefficient of Half/Corollary
Jump to navigation
Jump to search
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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $53 \ \text{(b)}$