Theorem: Pfaff-Saalschütz Theorem

:$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } { c^{\overline n} \paren {c - a - b}^{\overline n} }$

Proof:
By definition of the generalized hypergeometric function:
:$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} } { c^{\overline k} \paren {1 + a + b - c - n}^{\overline k} } \dfrac {1^k} {k!}$
From Euler's Transformation, we have:
{{begin-eqn}}
{{eqn | l = \map F {a, b; c; x}
      | r = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x }
      | c = Euler's Transformation
}}
{{eqn | n = 1
      | l = \paren {1 - x}^{a + b - c} \map F {a, b; c; x}
      | r = \map F {c - a, c - b; c; x }
      | c = multiplying both sides by $\paren {1 - x}^{a + b - c}$
}}
{{end-eqn}}
Next, we observe that:
{{begin-eqn}}
{{eqn | l = \paren {1 - x}^{a + b - c}
      | r = \sum_{k \mathop = 0}^\infty \binom {a + b - c} k \paren {-x}^k
      | c = Binomial Theorem: Complex Numbers
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \paren {\binom {c - a - b + k - 1} k \paren {-1}^k} \paren {-x}^k
      | c = Negated Upper Index of Binomial Coefficient
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \binom {c - a - b + k - 1} k x^k
      | c = $\paren {-1}^{2 k} = 1$
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {c - a - b + k - 1}!} {k! \paren {c - a - b - 1}! } x^k
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {c - a - b}^{\overline k} } {k!} x^k
      | c = Rising Factorial as Quotient of Factorials
}}
{{end-eqn}}
Equating coefficients of $x^n$ in $(1)$ above, we obtain:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \dfrac {\paren {c - a - b}^{\overline {n - k} } } {\paren {n - k}!} \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {c - a - b - 1 + n - k}!} {\paren {c - a - b - 1}!} \times \dfrac {\paren {n - k}!} {\paren {n - k}!} } \paren {\dfrac 1 {\paren {n - k}!} \times \dfrac {n!} {n!} } \dfrac {a^{\overline k} b^{\overline k} } { k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Rising Factorial as Quotient of Factorials and multiplying by $1$
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dbinom {c - a - b - 1 + n - k} {n - k} \paren {n - k}!} \paren {\dfrac {n^{\underline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = {{Defof|Binomial Coefficient}} and Falling Factorial as Quotient of Factorials
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dbinom {-c + a + b} {n - k} \paren {n - k}! \paren {-1}^{n - k} } \paren {\dfrac {\paren {-1}^k \paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Negated Upper Index of Binomial Coefficient and Rising Factorial in terms of Falling Factorial of Negative
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dbinom {-c + a + b} {n - k} \paren {n - k}! \paren {-1}^n} \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Product of Powers $\paren {-1}^{n - k} \times \paren {-1}^k = \paren {-1}^n$
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {a + b - c}!} {\paren {n - k}! \paren {a + b - c - n + k}! } \paren {n - k}! \paren {-1}^n } \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {a + b - c}!} {\paren {a + b - c - n + k}! } \paren {-1}^n } \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac { a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = $\paren {n - k}!$ cancels
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {a + b - c}!} {\paren {a + b - c - n + k}! } \times \dfrac {\paren {a + b - c - n}!} {\paren {a + b - c - n}!} \paren {-1}^n } \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac { a^{\overline k} b^{\overline k} } { k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = multiplying by $1$
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {a + b - c}!} {\paren {1 + a + b - c - n}^{\overline k} } \times \dfrac 1 {\paren {a + b - c - n}!} \paren {-1}^n} \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Rising Factorial as Quotient of Factorials
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {a + b - c}^{\underline n} } {\paren {1 + a + b - c - n}^{\overline k} } \paren {-1}^n} \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = {{Defof|Falling Factorial}}
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \paren {\dfrac {\paren {c - a - b}^{\overline n} \paren {-1}^n } {\paren {1 + a + b - c - n}^{\overline k} } \paren {-1}^n } \paren {\dfrac {\paren {-n}^{\overline k} } {n!} } \dfrac {a^{\overline k} b^{\overline k} } {k! c^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Rising Factorial in terms of Falling Factorial of Negative
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \dfrac {a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} \paren {c - a - b}^{\overline n} } {n! k! c^{\overline k} \paren {1 + a + b - c - n}^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }
      | c = Product of Powers $\paren {-1}^n \times \paren {-1}^n = \paren {-1}^{2 n} = 1$ and rearranging
}}
{{eqn | ll= \leadsto
      | l = \sum_{k \mathop = 0}^n \dfrac {a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} } {k! c^{\overline k} \paren {1 + a + b - c - n}^{\overline k} }
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {c^{\overline n} \paren {c - a - b}^{\overline n} }
      | c = multiplying both sides by $\dfrac {n!} {\paren {c - a - b}^{\overline n} }$
}}
{{eqn | ll= \leadsto
      | l = \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1}
      | r = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {c^{\overline n} \paren {c - a - b}^{\overline n} }
      | c = {{Defof|Generalized Hypergeometric Function}}
}}
{{end-eqn}}
{{qed}}