Theorem: Solution to Hypergeometric Differential Equation

The '''hypergeometric series''':
{{begin-eqn}}
{{eqn | l = \map F {a, b, c; z}
      | r = \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} z^n
}}
{{end-eqn}}
defines a solution to the '''hypergeometric differential equation''':
:$x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \dfrac {\d y} {\d x} - a b y = 0$

Proof:
Let:
{{begin-eqn}}
{{eqn | l = y
      | r = 1 + \frac {a b} {1! c} x + \frac {a \paren {a + 1} b \paren {b + 1} } {2! c \paren {c + 1} } x^2 + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! c \paren {c + 1} \paren {c + 2} } x^3 + \cdots
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n
      | c = 
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {\d y} {\d x}
      | r = \frac {a b} { \, c} + \dfrac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } x + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {2! c \paren {c + 1} \paren {c + 2} } x^2 + \cdots
      | c = Derivative of Constant and Power Rule for Derivatives
}}
{{eqn | r = \frac {a b} c \sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
      | c = 
}}
{{end-eqn}}
And:
{{begin-eqn}}
{{eqn | l = \frac {\d^2 y} {\d x^2}
      | r = \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {2! c \paren {c + 1} \paren {c + 2} } x + \cdots
      | c = Derivative of Constant and Power Rule for Derivatives
}}
{{eqn | r = \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n}  n!} x^n
      | c = 
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = 0
      | r = x \paren {1 - x} \frac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \frac {\d y} {\d x} - a b y
      | c =
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = x \frac {a \paren {a + 1} b \paren {b + 1} } { \, c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n - x^2 \frac {a \paren {a + 1} b \paren {b + 1} } { \, c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n
      | c = substituting from above
}}
{{eqn | o = 
      | ro= +
      | r = c \frac {a b} c \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
      | c = substituting from above
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} x \frac {a b} c \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
      | c = substituting from above
}}
{{eqn | o = 
      | ro= -
      | r = a b \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n
      | c = substituting from above
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = extracting first term
}}
{{eqn | o = 
      | ro= +
      | r = c \frac {a b} c \paren {1 + \frac {\paren {a + 1} \paren {b + 1} } {\paren {c + 1} } x + \sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }
      | c = extracting first $2$ terms
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \frac {a b} c \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }
      | c = extracting first term
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {1 + \frac {a b} c x + \sum_{n \mathop = 2}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n }
      | c = extracting first term
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = c \dfrac {a b} c \paren {\frac {\paren {a + 1} \paren {b + 1} } {\paren {c + 1} } x + \sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }
      | c = $a b$ cancels
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \frac {a b} { \, c} \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\frac {a b} c x + \sum_{n \mathop = 2}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n }
      | c = $a b$ cancels
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 1}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = $x$ term cancels
}}
{{eqn | o = 
      | ro= +
      | r = c \dfrac {a b} c \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }
      | c = $x$ term cancels
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \frac {a b} { \, c} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }
      | c = $x$ term cancels
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 2}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} x^n }
      | c = $x$ term cancels
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
}}
{{eqn | o = 
      | ro= +
      | r = c \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 2}^{\overline {n - 1} } \paren {b + 2}^{\overline {n - 1} } } {\paren {c + 2}^{\overline {n - 1} } n!} x^n }
      | c = divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline {n - 1} } \paren {b + 2}^{\overline {n - 1} } } {\paren {c + 2}^{\overline {n - 1} } n!} x^{n + 1} }
      | c = divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 2}^{\overline {n - 2} } \paren {b + 2}^{\overline {n - 2} } } {\paren {c + 2}^{\overline {n - 2} } n!} x^n }
      | c = divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 1}!} x^{n + 2} } - \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = reindex sum
}}
{{eqn | o = 
      | ro= +
      | r = c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 2}!} x^{n + 2} }
      | c = reindex sum
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} x^{n + 2} }
      | c = reindex sum
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }
      | c = reindex sum
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 1}!} x^{n + 2} } + c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 2}!} x^{n + 2} }
      | c = rearranging
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }
      | c = rearranging
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} x^{n + 2} }
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} \frac {\paren {n + 2} } {\paren {n + 2} } \frac {\paren {a + 2 + n} \paren {b + 2 + n} } {\paren {c + 2 + n} } x^{n + 2} } + c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \frac {\paren {a + 2 + n} \paren {b + 2 + n} } {\paren {c + 2 + n} } x^{n + 2} }
      | c = Sum of Indices of Rising Factorial and multiplying top and bottom by $\paren {n + 2}$ 
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} \frac {\paren {n + 1} \paren {n + 2} } {\paren {n + 1} \paren {n + 2} } x^{n + 2} }
      | c = Sum of Indices of Rising Factorial and multiplying top and bottom by $\paren {n + 1} \paren {n + 2}$ 
}}
{{eqn | o = 
      | ro= -
      | r = \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} \frac {\paren {n + 2} } {\paren {n + 2} } x^{n + 2} }
      | c = multiplying top and bottom by $\paren {n + 2}$ 
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \frac {\paren {a + 2 + n} \paren {b + 2 + n} \paren {n + 2 + c} } {\paren {c + 2 + n} } x^{n + 2} }
      | c = Linear Combination of Convergent Series and {{Defof|Factorial}}
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!}  \paren {n + 1} \paren {n + 2} x^{n + 2} }
      | c = {{Defof|Factorial}}
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {n + 2} \paren {a + b + 1} x^{n + 2} }
      | c = {{Defof|Factorial}}
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {a + 2 + n} \paren {b + 2 + n} x^{n + 2} }
      | c = $\paren {n + 2 + c}$ cancels
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!}  \paren {n + 1} \paren {n + 2} x^{n + 2} }
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {n + 2} \paren {a + b + 1} x^{n + 2} }
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 0
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {a b + 2 a + n a + 2 b + 4 + 2 n + n b + 2 n + n^2 } x^{n + 2} }
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!}  \paren {n^2 + 3 n + 2 + n a + n b + n + 2 a + 2 b + 2 + a b} x^{n + 2} }
      | c = 
}}
{{end-eqn}}
{{qed}}