Theorem: Hypergeometric Function Difference Equation/Formulation 2

{{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x} - \map { {}_1 \operatorname F_1} {a + 1; b; x}
      | r = \frac {\paren {a - b + 1} x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}}

Proof:
{{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
      | r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} } {\paren {b - 1}^{\overline n} n!} x^n
      | c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cdots + \frac {\map \Gamma {a + n} \map \Gamma {b - 1} } {\map \Gamma {b - 1 + n} \map \Gamma {a } } \frac {x^n} {n!} + \cdots
      | c = {{Defof|Rising Factorial}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cdots + \frac {\map \Gamma {a + n} \map \Gamma {b - 1} } { \map \Gamma {b - 1 + n} \map \Gamma a } \times \frac {\paren {b - 1} \paren {b - 1 + n} } {\paren {b - 1} \paren {b - 1 + n} }  \times \frac {x^n} {n!} + \cdots
      | c = multiplying by $1$
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cdots + \frac {\paren {b - 1 + n} } {\paren {b - 1} } \frac {\map \Gamma {a + n} \map \Gamma b } {\map \Gamma {b + n} \map \Gamma a } \frac {x^n} {n!} + \cdots
      | c = Gamma Difference Equation
}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a + 1; b; x}
      | r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} } {b^{\overline n} n!} x^n
      | c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} } b x + \cdots + \frac {\map \Gamma {a + 1 + n} \map \Gamma {b } } {\map \Gamma {b + n} \map \Gamma {a + 1 } } \frac {x^n} {n!} + \cdots
      | c = {{Defof|Rising Factorial}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} } b x + \cdots + \frac {\paren {a + n} } a \frac {\map \Gamma {a + n} \map \Gamma {b } } {\map \Gamma {b + n} \map \Gamma a } \frac {x^n} {n!} + \cdots
      | c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
      | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
      | r = 1 + \frac a {\paren {b - 1} } x + \cdots + \frac {\paren {b - 1 + n} } {\paren {b - 1} } \frac {\map \Gamma {a + n} \map \Gamma b } {\map \Gamma {b + n} \map \Gamma a } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | lo = -
      | l = \map { {}_1 \operatorname F_1} {a + 1; b; x}
      | r = 1 + \frac {\paren {a + 1} } b x + \cdots + \frac {\paren {a + n} } a \frac {\map \Gamma {a + n} \map \Gamma {b } } {\map \Gamma {b + n} \map \Gamma a } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \paren { \frac a {b - 1} - \frac {\paren {a + 1} } b  } x + \cdots + \paren { 1 + \frac n {\paren {b - 1} } - 1 - \frac n a } \frac {\map \Gamma {a + n} \map \Gamma b } {\map \Gamma {b + n} \map \Gamma {a } } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \frac {\paren {a - b + 1} } {b \paren {b - 1} } x + \cdots + \paren { \frac {n \paren {a - b + 1} } {a \paren {b - 1} } } \frac {\map \Gamma {a + n} \map \Gamma b } {\map \Gamma {b + n} \map \Gamma {a } } \frac {x^n} {n!} + \cdots
      | c = 
}}
{{eqn | r = \frac {\paren {a - b + 1} } {b \paren {b - 1} } x + \cdots + \paren { \frac {n \paren {a - b + 1} } {a \paren {b - 1} } } \frac {\map \Gamma {a + n} \map \Gamma {b } } {\map \Gamma {b + n} \map \Gamma {a } } \times \frac b b \times x \times \frac {x^{n - 1} } {n!} + \cdots
      | c = multiplying by $1$ and factoring out an $x$
}}
{{eqn | r = \frac {\paren {a - b + 1} } {b \paren {b - 1} } x + \cdots + \frac {\paren {a - b + 1} } {b \paren {b - 1} } \frac {\map \Gamma {a + n} \map \Gamma {b + 1 } } {\map \Gamma {b + n} \map \Gamma {a + 1 } } \frac {x^{n - 1} } {\paren {n - 1}!} + \cdots
      | c = Gamma Difference Equation and canceling an $n$
}}
{{eqn | r = \frac {\paren {a - b + 1} } {b \paren {b - 1} } x + \cdots + \frac {\paren {a - b + 1} } {b \paren {b - 1} } \frac {\map \Gamma {a + 1 + n} \map \Gamma {b + 1 } } {\map \Gamma {b + 1 + n} \map \Gamma {a + 1 } } \frac {x^n } {n!} + \cdots
      | c = $n \to n + 1$
}}
{{eqn | r = \frac {\paren {a - b + 1} x} {b \paren {b - 1} }  \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} } {\paren {b + 1}^{\overline n} n!} x^n
      | c = 
}}
{{eqn | r = \frac {\paren {a - b + 1} x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
      | c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{end-eqn}}
{{qed}}