Theorem: 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} } $

Proof:
From Gauss's Hypergeometric Theorem, we have:
:$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$
Therefore:
{{begin-eqn}}
{{eqn | l = \map F {b + n, c + n; 1 + a + 2 n; 1}
      | r = \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} - \paren {c + n} } } {\map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} } \map \Gamma {\paren {1 + a + 2 n} - \paren {c + n} } }
      | c = Gauss's Hypergeometric Theorem
}}
{{eqn | n = 1
      | r = \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a + n - b } \map \Gamma {1 + a + n - c } }
      | c = simplifying
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | o = 
      | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {1 + a - b} \map \Gamma {1 + a - c} } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {1 + a - b, 1 + a - c} } \, \middle \vert \, 1}
      | c = 
}}
{{eqn | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {1 + a - b} \map \Gamma {1 + a - c} } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} \dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {1 + a - b + n} } {\map \Gamma {1 + a - b} } \dfrac {\map \Gamma {1 + a - c + n} } {\map \Gamma {1 + a - c} } } \dfrac {1^n} {n!}
      | c = {{Defof|Hypergeometric Function}} and Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a - b + n} \map \Gamma {1 + a - c + n} } \times \dfrac {\map \Gamma {1 + a + 2 n} } {\map \Gamma {1 + a + 2 n} } \times \dfrac {\map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a - b - c} }
      | c = multiplying by $1$ and $1^n = 1$
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \map F {b + n, c + n; 1 + a + 2 n; 1}
      | c = from $(1)$ above
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + n + m} } {\map \Gamma {b + n} } \dfrac {\map \Gamma {c + n + m} } {\map \Gamma {c + n} } } {\dfrac {\map \Gamma {1 + a + 2 n + m} } {\map \Gamma {1 + a + 2 n} } } \dfrac {1^m} {m!}
      | c = {{Defof|Hypergeometric Function}} and Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n + m} \map \Gamma {c + n + m} } {n! m! \map \Gamma {1 + a + 2 n + m} \map \Gamma {- b - c + a + 1} }
      | c = simplifying
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} }
      | c = letting $p = m + n$, so $m = p - n$ and Product of Absolutely Convergent Series
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} } \times \dfrac {\map \Gamma {1 + a + p} } {\map \Gamma {1 + a + p} } \times \dfrac {\map \Gamma a } {\map \Gamma a } \times \dfrac {p!} {p!} \times \paren {-1}^n \times \paren {-1}^n
      | c = multiplying by $1$
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \sum_{n \mathop = 0}^p \dfrac {\paren {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} } \paren {\dfrac {\paren {-1}^n p!} {\paren {p - n}!} } } {\dfrac {\map \Gamma {1 + a + n + p} } {\map \Gamma {1 + a + p} } } \dfrac {\paren {-1}^n} {n!}
      | c = rearranging terms
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \map F {a, -p; 1 + a + p; -1}
      | c = {{Defof|Hypergeometric Function}}, Rising Factorial in terms of Falling Factorial of Negative and Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {p + a + 1} \map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }
      | c = Kummer's Hypergeometric Theorem
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }
      | c = canceling $\map \Gamma {1 + a + p}$
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} } \times \dfrac {\map \Gamma b} {\map \Gamma b} \times \dfrac {\map \Gamma c} {\map \Gamma c}
      | c = multiplying by $1$
}}
{{eqn | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \sum_{p \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + p} } {\map \Gamma b } \dfrac {\map \Gamma {c + p} } {\map \Gamma c  } } {\dfrac {\map \Gamma {p + \dfrac a 2 + 1} } {\map \Gamma {\dfrac a 2 + 1} } } \dfrac 1 {p!}
      | c = 
}}
{{eqn | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \map F {b, c; \dfrac a 2 + 1; 1}
      | c = {{Defof|Hypergeometric Function}} and Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }
      | c = Gauss's Hypergeometric Theorem
}}
{{end-eqn}}
We now have:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {- b + a + 1, - c+ a + 1} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {- b + a + 1, - c + a + 1} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }
      | c = 
}}
{{end-eqn}}
Substituting $a = n$, $b = -x$ and $c = -y$, we obtain:
{{begin-eqn}}
{{eqn | l = \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {1 + a - b, 1 + a - c} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b + \dfrac a 2 + 1} \map \Gamma {-c + \dfrac a 2 + 1} }
      | c = before substitution
}}
{{eqn | ll= \leadsto
      | 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 = after substitution
}}
{{end-eqn}}
{{qed}}