Theorem: Gaussian Hypergeometric Function Difference Equation/Formulation 1

{{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x} - \map F {a, b, c; x}
      | r = \frac {a \paren {c - b} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
}}
{{end-eqn}}

Proof:
{{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x}
      | r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \frac {\map \Gamma {a + n} \map \Gamma {b + 1 + n} \map \Gamma {c + 1} } {\map \Gamma {c + 1 + n} \map \Gamma {a } \map \Gamma {b + 1} } \frac {x^n} {n!} + \cdots
      | c = {{Defof|Rising Factorial}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \frac {\map \Gamma {a + n} \paren {b + n} \map \Gamma {b + n} c \map \Gamma c } {\paren {c + n} \map \Gamma {c + n} \map \Gamma a b \map \Gamma b } \frac {x^n} {n!} + \cdots
      | c = Gamma Difference Equation
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \frac {c \paren {b + n} } {b \paren {c + n} } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma c } {\map \Gamma {c + n} \map \Gamma a \map \Gamma b } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | l = \map F {a, b, c; x}
      | r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a b } {c } x + \cdots + \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + n} \map \Gamma {a } \map \Gamma {b } } \frac {x^n} {n!} + \cdots
      | c = {{Defof|Rising Factorial}}
}}
{{eqn | ll= \leadsto
      | l = \map F {a, b + 1, c + 1; x}
      | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \frac {c \paren {b + n} } {b \paren {c + n} } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma c } {\map \Gamma {c + n} \map \Gamma a \map \Gamma b } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | lo = -
      | l = \map F {a, b, c; x}
      | r = 1 + \frac {a b } {c } x + \cdots + \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + n} \map \Gamma {a } \map \Gamma {b } } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \paren { \frac {\paren {b + 1} } {\paren {c + 1} } - \frac b c } a x + \cdots + \paren { \frac {c \paren {b + n} } {b \paren {c + n} } - 1} \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + n} \map \Gamma {a } \map \Gamma {b } } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { \frac {n \paren {c - b} } {b \paren {c + n} } } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + n} \map \Gamma {a } \map \Gamma {b } } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { \frac {n \paren {c - b} } {b \paren {c + n} } } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + n} \map \Gamma {a } \map \Gamma {b } } \times \frac a a \times x \times \frac {x^{n - 1} } {n!} + \cdots
      | c = multiplying by $1$ and factoring out an $x$
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { a \paren {c - b} x  } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + 1 + n} \map \Gamma {a + 1 } \map \Gamma {b + 1 } } \frac {x^{n - 1} } {\paren {n - 1}!} + \cdots
      | c = Gamma Difference Equation and canceling an $n$
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { a \paren {c - b} x  } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c } } {\map \Gamma {c + 1 + n} \map \Gamma {a + 1 } \map \Gamma {b + 1 } } \times \frac {c \paren {c + 1} } {c \paren {c + 1} } \times \frac {x^{n - 1} } {\paren {n - 1}!} + \cdots
      | c = multiplying by $1$
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { \frac {a \paren {c - b} x } {c \paren {c + 1} }  } \frac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + 2 } } {\map \Gamma {c + 1 + n} \map \Gamma {a + 1 } \map \Gamma {b + 1 } } \frac {x^{n - 1} } {\paren {n - 1}!} + \cdots
      | c = Gamma Difference Equation
}}
{{eqn | r = \paren { \frac {a \paren {c - b} } {c \paren {c + 1} } } x + \cdots + \paren { \frac {a \paren {c - b} x } {c \paren {c + 1} }  } \frac {\map \Gamma {a + 1 + n} \map \Gamma {b + 1 + n} \map \Gamma {c + 2 } } {\map \Gamma {c + 2 + n} \map \Gamma {a + 1 } \map \Gamma {b + 1 } } \frac {x^n } {n!} + \cdots
      | c = $n \to n + 1$
}}
{{eqn | r = \frac {a \paren {c - b} x } {c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n
      | c = 
}}
{{eqn | r = \frac {a \paren {c - b} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
      | c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{end-eqn}}
{{qed}}