Theorem: Kummer's Hypergeometric Theorem

:$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$

Proof:
First we note the definition of Gaussian hypergeometric function:
:$\map F {n, -x; x + n + 1; -1} = \ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}$
where $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
Two lemmata:
=== Lemma 1 ===
{{:Kummer's Hypergeometric Theorem/Lemma 1}}{{qed|lemma}}
=== Lemma 2 ===
{{:Kummer's Hypergeometric Theorem/Lemma 2}}{{qed|lemma}}
We use Dixon's Hypergeometric Theorem:
:$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }$
where $\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$.
So:
{{begin-eqn}}
{{eqn | l = \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }
      | c = Dixon's Hypergeometric Theorem
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }
      | c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {\paren {-y}^{\overline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\map \Gamma {y + n + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} }
      | c = reorganizing both sides: isolating $y$
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren{-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\map \Gamma {y + n + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} }
      | c = on the {{LHS}}: Rising Factorial in terms of Falling Factorial of Negative
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren{-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} }
      | c = on the {{RHS}}: Rising Factorial as Quotient of Factorials
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^\infty \dfrac {n^{\overline k} \paren{-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \paren 1 \dfrac {\paren {-1}^k} {k!}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times 1
      | c = {{Lemma|Kummer's Hypergeometric Theorem|1}} and {{Lemma|Kummer's Hypergeometric Theorem|2}}: letting $y \to \infty$
}}
{{eqn | ll= \leadsto
      | l = \map F {n, -x; x + n + 1; -1}
      | r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{end-eqn}}
{{qed}}