Chu-Vandermonde Identity/Proof 2

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Theorem

$\ds \sum_{k \mathop = 0}^n \binom r k \binom s {n - k} = \binom {r + s} n$


Proof

The Chu-Vandermonde Identity is a special case of Gauss's Hypergeometric Theorem:

$\map { {}_2F_1} {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

when:

$\map \Re {c - a - b} \gt 0$

where:

$\map { {}_2F_1} {a, b; c; 1}$ is the hypergeometric series: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


Starting on the left hand side:

\(\ds \map { {}_2F_1} {-n, b; c; 1}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {-n}^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}\) Definition of Hypergeometric Function
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {-n}^{\overline k} b^{\overline k} } { k! c^{\overline k} }\) $1^k = 1$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {-1}^k \paren {n}^{\underline k} b^{\overline k} } { k! c^{\overline k} }\) Rising Factorial in terms of Falling Factorial of Negative
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {-1}^k n! b^{\overline k} } {k! \paren {n - k}! c^{\overline k} }\) Falling Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} } {c^{\overline k} }\) Definition of Binomial Coefficient, $\dbinom n k = 0$ when $k > n$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} } {c^{\overline k} } \dfrac {c^{\overline n} } { c^{\overline n} }\) multiplying by $1$
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} c^{\overline {n} } } {c^{\overline {k} } }\) moving $\dfrac 1 { c^{\overline n} }$ outside the sum
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {\paren {b - 1 + k}!} {\paren {b - 1}!} \dfrac {\dfrac {\paren {c - 1 + n}!} {\paren {c - 1}!} } {\dfrac {\paren {c - 1 + k}!} {\paren {c - 1}!} }\) Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {\paren {b - 1 + k}!} {\paren {b - 1}!} \times \dfrac {k!} {k!} \times \dfrac {\paren {c - 1 + k + \paren {n - k} }!} {\paren {c - 1 + k}!}\) multiplying by $1$ and simplifying
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k k! \dbinom n k \dbinom {b - 1 + k} k \paren {c - 1 + k}^{\overline {n - k} }\) Definition of Binomial Coefficient and Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n k! \dbinom n k \dbinom {-b} k \paren {c - 1 + k}^{\overline {n - k} }\) Negated Upper Index of Binomial Coefficient
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n k! \dbinom n k \dfrac {-b!} {k! \paren {-b - k}!} \paren {c - 1 + k}^{\overline {n - k} }\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \dfrac {-b!} {\paren {-b - k}!} \paren {c - 1 + k}^{\overline {n - k} }\) $k!$ cancels
\(\ds \) \(=\) \(\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \paren {1 - b - k}^{\overline k} \paren {c - 1 + k}^{\overline {n - k} }\) Rising Factorial as Quotient of Factorials

Moving to the right hand side, we let $a = -n$:

\(\ds \map { {}_2F_1} {-n, b; c; 1}\) \(=\) \(\ds \dfrac {\map \Gamma c \map \Gamma {c - \paren {-n} - b} } {\map \Gamma {c - \paren {-n} } \map \Gamma {c - b} }\)
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \map \Gamma {c - b + n} } {\map \Gamma {c + n } \map \Gamma {c - b} }\)
\(\ds \) \(=\) \(\ds \dfrac {\dfrac {\map \Gamma {c - b + n} } {\map \Gamma {c - b} } } {\dfrac {\map \Gamma {c + n } } {\map \Gamma {c } } }\) rearranging
\(\ds \) \(=\) \(\ds \dfrac {\paren {c - b}^{\overline n} } {c^{\overline n} }\) Rising Factorial as Quotient of Factorials

Finally, setting the left hand side equal to the right hand side, we see the Chu-Vandermonde Identity:

$\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \paren {1 - b - k}^{\overline k} \paren {c - 1 + k}^{\overline {n - k} } = \dfrac {\paren {c - b}^{\overline n} } {c^{\overline n} }$

$\blacksquare$


Source of Name

This entry was named for Alexandre-Théophile Vandermonde and Chu Shih-Chieh.