Theorem: Pfaff's Transformation

:$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$

Proof:
From Euler's Integral Representation of Hypergeometric Function, we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - xt}^{- a} \rd t$
Letting $s = \paren {1 - t}$, we now have:
{{begin-eqn}}
{{eqn | l = \map F {a, b; c; x}
      | r = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_1^0 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x\paren {1 - s} }^{- a} \paren {-\rd s}
      | c = substituting $s = \paren {1 - t}$
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x\paren {1 - s} }^{- a} \rd s
      | c = reversing limits of integration
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x + xs }^{- a} \rd s
      | c = simplifying
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x + xs }^{- a} \rd s \times \dfrac {\paren {1 - x}^{-a} } {\paren {1 - x}^{-a} }
      | c = multiplying by $1$
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 + \dfrac {xs} {1 - x} }^{- a} \rd s
      | c = rearranging
}}
{{eqn | n = 1
      | ll= \leadsto
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - \dfrac {xs} {x - 1} }^{- a} \rd s
      | c = switching the sign in the last term
}}
{{end-eqn}}
Letting $\size x < \dfrac 1 2$ and expanding the product of $\paren {1 - \dfrac {xs} {x - 1} }^{-a}$:
{{begin-eqn}}
{{eqn | l = \paren {1 - \dfrac {xs} {x - 1} }^{-a}
      | r = \sum_{k \mathop = 0}^\infty \binom {-a} k \paren {-1}^k \paren {\dfrac {xs} {x - 1} }^k
      | c = Binomial Theorem - Complex Numbers
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \paren {\binom {a + k - 1} k \paren {-1}^k} \paren {-1}^k \paren {\dfrac {xs} {x - 1} }^k
      | c = Negated Upper Index of Binomial Coefficient
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \binom {a + k - 1} k \paren {\dfrac {xs} {x - 1} }^k
      | c = $\paren {-1}^{2k} = 1$
}}
{{eqn | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {a + k - 1}! } {k! \paren {a - 1}! } \paren {\dfrac x {\paren {x - 1} } }^k s^k
      | c = {{Defof|Binomial Coefficient}}
}}
{{eqn | n = 2
      | ll= \leadsto
      | r = \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k s^k
      | c = Rising Factorial as Quotient of Factorials
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - \dfrac {xs} {x - 1} }^{- a} \rd s
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k s^k \rd s
      | c = substituting $\paren {2}$ into $\paren {1}$
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} s^k \rd s
      | c = 
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \int_0^1 \paren {1 - s}^{b - 1} s^{k + c - b - 1} \rd s
      | c = Product of Powers
}}
{{eqn | ll= \leadsto
      | r = \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \dfrac {\map \Gamma {k + c - b} \map \Gamma b } {\map \Gamma {k + c} }
      | c = {{Defof|Beta Function}}
}}
{{eqn | ll= \leadsto
      | r = \paren {1 - x}^{-a} \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} \paren {c - b}^{\overline k} } { k! c^{\overline k} } \paren {\dfrac x {\paren {x - 1} } }^k
      | c = Rising Factorial as Quotient of Factorials and $\map \Gamma b$ cancels
}}
{{eqn | ll= \leadsto
      | r = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{end-eqn}}
Therefore:
:$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$
{{qed}}