Theorem: Thomae's Transformation

:$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1} = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1} $

Proof:
First, we observe that:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \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 {e + n} } {\map \Gamma e } \dfrac {\map \Gamma {f + n} } {\map \Gamma f } } \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 {e + n} \map \Gamma {f + n} } \times \dfrac {\map \Gamma {e + f - a + n} } {\map \Gamma {e + f - a + n} }
      | c = multiplying by $1$ and $1^n = 1$
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \times \dfrac {\map \Gamma {e + f - a + n} \map \Gamma {a + n} } {\map \Gamma {e + n} \map \Gamma {f + n} }
      | c = rearranging
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \times \dfrac {\map \Gamma {e + f - a + n} \map \Gamma {\paren {e + f - a + n} - \paren {e - a} - \paren {f - a} } } {\map \Gamma {\paren {e + f - a + n} - \paren {f - a} } \map \Gamma {\paren {e + f - a + n} - \paren {e - a} } }
      | c = rearranging further
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \map F {e - a, f - a; e + f - a + n; 1}
      | c = Gauss's Hypergeometric Theorem
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {e - a + m} } {\map \Gamma {e - a } } \dfrac {\map \Gamma {f - a + m} } {\map \Gamma {f - a } } } {\dfrac {\map \Gamma {e + f - a + n + m} } {\map \Gamma {e + f - a + 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 {b + n} \map \Gamma {c + n} \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! n! \map \Gamma {e + f - a + n + m} \map \Gamma {e - a } \map \Gamma {f - a } }
      | c = simplifying: $1^m = 1$ and $\map \Gamma {e + f - a + n}$ cancels
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! n! \map \Gamma {e + f - a + n + m} \map \Gamma {e - a } \map \Gamma {f - a } } \times \dfrac {\map \Gamma b } {\map \Gamma b} \times \dfrac {\map \Gamma c } {\map \Gamma c} \times \dfrac {\map \Gamma {e + f - a + m} } {\map \Gamma {e + f - a + m} }
      | c = multiplying by $1$
}}
{{eqn | r = \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {e + f - a + n + m} } {\map \Gamma {e + f - a + m} } } \dfrac {1^n} {n!}
      | c = rearranging
}}
{{eqn | r = \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \map F {b, c; e + f - a + m; 1}
      | c = {{Defof|Hypergeometric Function}} and Rising Factorial as Quotient of Factorials
}}
{{eqn | r = \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {e + f - a + m} \map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} }
      | c = Gauss's Hypergeometric Theorem and $s = e + f - a - b - c$
}}
{{eqn | r = \map \Gamma b \map \Gamma c \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} }
      | c = rearranging and $\map \Gamma {e + f - a + m}$ cancels
}}
{{eqn | r = \map \Gamma b \map \Gamma c \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} } \times \dfrac {\map \Gamma s } {\map \Gamma s } \times \dfrac {\map \Gamma {s + b} } {\map \Gamma {s + b} } \times \dfrac {\map \Gamma {s + c} } {\map \Gamma {s + c} }
      | c = multiplying by $1$
}}
{{eqn | r = \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {e - a + m} } {\map \Gamma {e - a } } \dfrac {\map \Gamma {f - a + m} } {\map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma s } } {\dfrac {\map \Gamma {s + b + m} } {\map \Gamma {s + b } } \dfrac {\map \Gamma {s + c + m} } {\map \Gamma {s + c } }  } \dfrac {1^m} {m!}
      | c = rearranging
}}
{{eqn | r = \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}
      | c = {{Defof|Hypergeometric Function}} and Rising Factorial as Quotient of Factorials
}}
{{end-eqn}}
We now have:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
      | r = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}
      | c = 
}}
{{end-eqn}}
{{qed}}