Theorem: Gauss's Continued Fraction

:$\ds \frac {\map F {\alpha, \beta + 1; \gamma + 1; x} } {\map F {\alpha, \beta; \gamma; x} } = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - \cfrac {d x} {1 - \cfrac {\ddots} {\ddots } } } } } }$
where:
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
:<nowiki>$\begin{array}{rc|rc}
a = & \dfrac {\alpha \paren {\gamma - \beta} } {\gamma \paren {\gamma + 1} } & b = & \dfrac {\paren {\beta + 1} \paren {\gamma - \alpha + 1} } {\paren {\gamma + 1} \paren {\gamma + 2} } \\
c = & \dfrac {\paren {\alpha + 1} \paren {\gamma - \beta + 1} } {\paren {\gamma + 2} \paren {\gamma + 3} } & d =  & \dfrac {\paren {\beta + 2} \paren {\gamma - \alpha + 2} } {\paren {\gamma + 3} \paren {\gamma + 4} } \\
e = & \dfrac {\paren {\alpha + 2} \paren {\gamma - \beta + 2} } {\paren {\gamma + 4} \paren {\gamma + 5} } & f =  & \dfrac {\paren {\beta + 3} \paren {\gamma - \alpha + 3} } {\paren {\gamma + 5} \paren {\gamma + 6} } \\
g = & \dfrac {\paren {\alpha + 3} \paren {\gamma - \beta + 3} } {\paren {\gamma + 6} \paren {\gamma + 7} } & h =  & \dfrac {\paren {\beta + 4} \paren {\gamma - \alpha + 4} } {\paren {\gamma + 7} \paren {\gamma + 8} } \\
\end{array}$</nowiki>

Proof:
:First, we demonstate:
:$\ds \dfrac {\map F {\alpha, \beta + 1, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} } = \dfrac 1 {1 - \dfrac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }$
{{begin-eqn}}
{{eqn | l = \map F {\alpha, \beta + 1, \gamma + 1; x} - \map F {\alpha, \beta, \gamma; x}
      | r = \frac {\alpha \paren {\gamma - \beta} x } {c \paren {\gamma + 1} } \map F {\alpha + 1, \beta + 1, \gamma + 2; x}
      | c = Gaussian Hypergeometric Function Difference Equation/Formulation 1
}}
{{eqn | ll = \leadsto
      | l = \frac {\map F {\alpha, \beta + 1, \gamma + 1; x} - \map F {\alpha, \beta, \gamma; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | r = \frac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | c = dividing both sides by $\map F {\alpha, \beta + 1, \gamma + 1; x}$
}}
{{eqn | ll = \leadsto
      | l = 1 - \frac {\map F {\alpha, \beta, \gamma; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | r = \frac {\alpha \paren {\gamma - \beta} x } {c \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \frac {\map F {\alpha, \beta, \gamma; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | r = 1 - \frac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} }
      | c = 
}}
{{eqn | ll = \leadsto
      | n = 1
      | l = \frac {\map F {\alpha, \beta + 1, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} }
      | r = \frac 1 {1 - \dfrac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }
      | c = taking the reciprocal of both sides
}}
{{end-eqn}}
:Next, we demonstate:
:$\ds \dfrac {\map F {\alpha + 1, \beta, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} } = \dfrac 1 {1 - \dfrac {\beta \paren {\gamma - \alpha} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} } }$
{{begin-eqn}}
{{eqn | l = \map F {\alpha + 1, \beta, \gamma + 1; x} - \map F {\alpha, \beta, \gamma; x}
      | r = \frac {\beta \paren {\gamma - \alpha} x } {\gamma \paren {\gamma + 1} } \map F {\alpha + 1, \beta + 1, \gamma + 2; x}
      | c = Gaussian Hypergeometric Function Difference Equation/Formulation 2
}}
{{eqn | ll = \leadsto
      | l = \frac {\map F {\alpha + 1, \beta, \gamma + 1; x} - \map F {\alpha, \beta, \gamma; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | r = \frac {\beta \paren {\gamma - \alpha} x } {\gamma \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | c = dividing both sides by $\map F {\alpha + 1, \beta, \gamma + 1; x}$
}}
{{eqn | ll = \leadsto
      | l = 1 - \frac {\map F {\alpha, \beta, \gamma; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | r = \frac {\beta \paren {\gamma - \alpha} x } {c \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \frac {\map F {\alpha, \beta, \gamma; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | r = 1 - \frac {\beta \paren {\gamma - \alpha} x } {\gamma \paren {\gamma + 1} } \frac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} }
      | c = 
}}
{{eqn | ll = \leadsto
      | n = 2
      | l = \frac {\map F {\alpha + 1, \beta, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} }
      | r = \frac 1 {1 - \dfrac {\beta \paren {\gamma - \alpha} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha + 1, \beta, \gamma + 1; x} } }
      | c = taking the reciprocal of both sides
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \frac {\map F {\alpha, \beta + 1, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} }
      | r = \frac 1 {1 - \dfrac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }
      | c = from $\paren {1 }$ above
}}
{{eqn | ll = \leadsto
      | r = \frac 1 {1 - a x \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }
      | c = $a = \dfrac {\alpha \paren {\gamma - \beta} } {\gamma \paren {\gamma + 1} }$
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \dfrac {\paren {\beta + 1} \paren {\gamma - \alpha + 1} x } {\paren {\gamma + 1} \paren {\gamma + 2} } \dfrac {\map F {\alpha + 1, \beta + 2, \gamma + 3; x} } {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } } }
      | c = from $\paren {2 }$ above
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - b x \dfrac {\map F {\alpha + 1, \beta + 2, \gamma + 3; x} } {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } } }
      | c = $b = \dfrac {\paren {\beta + 1} \paren {\gamma - \alpha + 1} } {\paren {\gamma + 1} \paren {\gamma + 2} }$
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \dfrac {\paren {\alpha + 1} \paren {\gamma - \beta + 1} x } {\paren {\gamma + 2} \paren {\gamma + 3} } \dfrac {\map F {\alpha + 2, \beta + 2, \gamma + 4; x} } {\map F {\alpha + 1, \beta + 2, \gamma + 3; x} } }  } }
      | c = from $\paren {1 }$ above
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - c x \dfrac {\map F {\alpha + 2, \beta + 2, \gamma + 4; x} } {\map F {\alpha + 1, \beta + 2, \gamma + 3; x} } }  } }
      | c = $c = \dfrac {\paren {\alpha + 1} \paren {\gamma - \beta + 1} } {\paren {\gamma + 2} \paren {\gamma + 3} }$
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - \dfrac {\paren {\beta + 2} \paren {\gamma - \alpha + 2} x } {\paren {\gamma + 3} \paren {\gamma + 4} } \dfrac {\map F {\alpha + 2, \beta + 3, \gamma + 5; x} } {\map F {\alpha + 2, \beta + 2, \gamma + 4; x} } } } } }
      | c = from $\paren {2 }$ above
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - d x \dfrac {\map F {\alpha + 2, \beta + 3, \gamma + 5; x} } {\map F {\alpha + 2, \beta + 2, \gamma + 4; x} } } } } }
      | c = $d = \dfrac {\paren {\beta + 2} \paren {\gamma - \alpha + 2} } {\paren {\gamma + 3} \paren {\gamma + 4} }$
}}
{{eqn | ll = \leadsto
      | r = \cdots
      | c = {{adinfinitum}}
}}
{{eqn | ll = \leadsto
      | r = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - \cfrac {d x} {1 - \cfrac {\ddots} {\ddots } } } } } }
      | c = 
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Carl Friedrich Gauss|cat = Gauss}}