Product of r Choose k with r Minus Half Choose k/Formulation 2

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Theorem

Let $k \in \Z$, $r \in \R$.

$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} {2 k} \dbinom {2 k} k} {4^k}$

where $\dbinom r k$ denotes a binomial coefficient.


Proof

From Binomial Coefficient expressed using Beta Function:

$(1): \quad \dbinom r k \dbinom {r - \frac 1 2} k = \dfrac 1 {\paren {r + 1} \map \Beta {k + 1, r - k + 1} \paren {r + \frac 1 2} \map \Beta {k + 1, r - k + \frac 1 2} }$


Then:

\(\ds \dbinom r {k + 1} \dbinom {r - \frac 1 2} {k + 1}\) \(=\) \(\ds \dfrac 1 {\paren {r + 1} \map \Beta {k + 2, r - k} \paren {r + \frac 1 2} \map \Beta {k + 2, r - k - \frac 1 2} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {r + 1} \frac {k + 1} {r + 1} \map \Beta {k + 1, r - k} \paren {r + \frac 1 2} \frac {k + 1} {r + \frac 1 2} \map \Beta {k + 1, r - k - \frac 1 2} }\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {r + 1} \frac {k + 1} {r + 1} \frac {r + 1} {r - k} \map \Beta {k + 1, r - k + 1} \paren {r + \frac 1 2} \frac {k + 1} {r + \frac 1 2} \frac {r + \frac 1 2} {r - k - \frac 1 2} \map \Beta {k + 1, r - k + \frac 1 2} }\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac {\paren {r - k} \paren {r - k - \frac 1 2} } {\paren {k + 1}^2} \times \frac 1 {\paren {r + 1} \map \Beta {k + 1, r - k + 1} \paren {r + \frac 1 2} \map \Beta {k + 1, r - k + \frac 1 2} }\) simplifying
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac {\paren {r - k} \paren {r - k - \frac 1 2} } {\paren {k + 1}^2} \dbinom r k \dbinom {r - \frac 1 2} k\) from $(1)$


Then:

\(\ds \dbinom {2 r} {2 k + 2} \dbinom {2 k + 2} {2 k + 2}\) \(=\) \(\ds \dfrac 1 {\paren {2 r + 1} \map \Beta {2 k + 3, 2 r - 2 k - 1} \paren {2 k + 3} \map \Beta {k + 2, k + 2} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {2 r + 1} \frac {2 k + 2} {2 r + 1} \map \Beta {2 k + 2, 2 r - 2 k - 1} \paren {2 k + 3} \frac {k + 1} {2 k + 3} \map \Beta {k + 1, k + 2} }\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {2 k + 2} \map \Beta {2 k + 2, 2 r - 2 k - 1} \paren {k + 1} \map \Beta {k + 1, k + 2} }\) simplification
\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {2 k + 2} \frac {2 k + 1} {2 r} \map \Beta {2 k + 1, 2 r - 2 k - 1} \paren {k + 1} \frac {k + 1} {2 k + 2} \map \Beta {k + 1, k + 1} }\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac {2 r} {\paren {k + 1}^2 \map \Beta {2 k + 1, 2 r - 2 k - 1} \paren {2 k + 1} \map \Beta {k + 1, k + 1} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {2 r} {\paren {k + 1}^2 \map \Beta {2 k + 1, 2 r - 2 k - 1} } \binom {2 k} k\) Binomial Coefficient expressed using Beta Function
\(\ds \) \(=\) \(\ds \dfrac {2 r} {\paren {k + 1}^2 \frac {2 r} {2 r - 2 k - 1} \map \Beta {2 k + 1, 2 r - 2 k} } \binom {2 k} k\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac {2 r - 2 k - 1} {\paren {k + 1}^2 \map \Beta {2 k + 1, 2 r - 2 k} } \binom {2 k} k\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {2 r - 2 k - 1} {\paren {k + 1}^2 \frac {2 r + 1} {2 r - 2 k} \map \Beta {2 k + 1, 2 r - 2 k + 1} } \binom {2 k} k\) Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$
\(\ds \) \(=\) \(\ds \dfrac {\paren {2 r - 2 k} \paren {2 r - 2 k - 1} } {\paren {k + 1}^2 \paren {2 r + 1} \map \Beta {2 k + 1, 2 r - 2 k + 1} } \binom {2 k} k\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\paren {2 r - 2 k} \paren {2 r - 2 k - 1} } {\paren {k + 1}^2} \dbinom {2 r} {2 k} \binom {2 k} k\) Binomial Coefficient expressed using Beta Function
\(\ds \) \(=\) \(\ds \dfrac {4 \paren {r - k} \paren {r - k - \frac 1 2} } {\paren {k + 1}^2} \dbinom {2 r} {2 k} \binom {2 k} k\)


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