Theorem: Chu-Vandermonde Identity

{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \binom r k \binom s {n - k}
      | r = \binom r 0 \binom s n + \binom r 1 \binom s {n - 1} + \binom r 2 \binom s {n - 2} + \cdots + \binom r n \binom s 0
      | c = 
}}
{{eqn | r = \binom {r + s} n
      | c = 
}}
{{end-eqn}}

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 {{LHS}}:
{{begin-eqn}}
{{eqn | l = \map { {}_2F_1} {-n, b; c; 1}
      | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-n}^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}
      | c = {{Defof|Hypergeometric Function}}
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-n}^{\overline k} b^{\overline k} } { k! c^{\overline k} }
      | c = $1^k = 1$
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k \paren n^{\underline k} b^{\overline k} } { k! c^{\overline k} }
      | c = Rising Factorial in terms of Falling Factorial of Negative
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k n! b^{\overline k} } {k! \paren {n - k}! c^{\overline k} }
      | c = Falling Factorial as Quotient of Factorials
}}
{{eqn | r = \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} } {c^{\overline k} }
      | c = {{Defof|Binomial Coefficient}}, $\dbinom n k = 0$ when $k > n$
}}
{{eqn | r = \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} }
      | c = multiplying by $1$
}}
{{eqn | r = \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} } 
      | c = moving $\dfrac 1 { c^{\overline n} }$ outside the sum
}}
{{eqn | r = \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}!} }
      | c = Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \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}!}
      | c = multiplying by $1$ and simplifying
}}
{{eqn | r = \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} } 
      | c = {{Defof|Binomial Coefficient}} and {{Defof|Rising Factorial}}
}}
{{eqn | r = \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n k! \dbinom n k \dbinom {-b} k \paren {c - 1 + k}^{\overline {n - k} } 
      | c = Negated Upper Index of Binomial Coefficient
}}
{{eqn | r = \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} } 
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \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} } 
      | c = $k!$ cancels
}}
{{eqn | r = \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} } 
      | c = Rising Factorial as Quotient of Factorials
}}
{{end-eqn}}
Moving to the {{RHS}}, we let $a = -n$:
{{begin-eqn}}
{{eqn | l = \map { {}_2F_1} {-n, b; c; 1}
      | r = \dfrac {\map \Gamma c \map \Gamma {c - \paren {-n} - b} } {\map \Gamma {c - \paren {-n} } \map \Gamma {c - b} }
      | c = 
}}
{{eqn | r = \dfrac {\map \Gamma c \map \Gamma {c - b + n} } {\map \Gamma {c + n} \map \Gamma {c - b} }
      | c = 
}}
{{eqn | r = \dfrac {\dfrac {\map \Gamma {c - b + n} } {\map \Gamma {c - b} } } {\dfrac {\map \Gamma {c + n} } {\map \Gamma c} }
      | c = rearranging
}}
{{eqn | r =  \dfrac {\paren {c - b}^{\overline n} } {c^{\overline n} }
      | c = Rising Factorial as Quotient of Factorials
}}
{{end-eqn}}
Finally, setting the {{LHS}} equal to the {{RHS}}, we see the Chu-Vandermonde Identity:
:$\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \paren {1 - b - k}^{\overline k} \paren {c - 1 + k}^{\overline {n - k} } = \dfrac {\paren {c - b}^{\overline n} } {c^{\overline n} }$
{{qed}}