Solution to Hypergeometric Differential Equation

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Theorem

The hypergeometric series:

\(\ds \map F {a, b, c; n}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} z^n\)

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:

\(\ds y\) \(=\) \(\ds 1 + \frac {a b} {1! c} x + \frac {a \paren {a + 1} b \paren {b + 1} } {2! c \paren {c + 1} } x^2 + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! c \paren {c + 1} \paren {c + 2} } x^3 + \cdots\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n\)

Then:

\(\ds \frac {\d y} {\d x}\) \(=\) \(\ds \frac {a b} { \, c} + \dfrac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } x + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {2! c \paren {c + 1} \paren {c + 2} } x^2 + \cdots\) Derivative of Constant and Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {a b} c \sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n\)

And:

\(\ds \frac {\d^2 y} {\d x^2}\) \(=\) \(\ds \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {2! c \paren {c + 1} \paren {c + 2} } x + \cdots\) Derivative of Constant and Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n\)

Therefore:

\(\ds 0\) \(=\) \(\ds x \paren {1 - x} \frac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \frac {\d y} {\d x} - a b y\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds x \frac {a \paren {a + 1} b \paren {b + 1} } { \, c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n - x^2 \frac {a \paren {a + 1} b \paren {b + 1} } { \, c \paren {c + 1} } \sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^n\) substituting from above
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \frac {a b} c \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n\) substituting from above
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} x \frac {a b} c \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n\) substituting from above
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n\) substituting from above
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\) extracting first term
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \frac {a b} c \paren {1 + \frac {\paren {a + 1} \paren {b + 1} } {\paren {c + 1} } x + \sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }\) extracting first $2$ terms
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \frac {a b} c \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }\) extracting first term
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {1 + \frac {a b} c x + \sum_{n \mathop = 2}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n }\) extracting first term
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \dfrac {a b} c \paren {\frac {\paren {a + 1} \paren {b + 1} } {\paren {c + 1} } x + \sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }\) $a b$ cancels
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \frac {a b} { \, c} \paren {x + \sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\frac {a b} c x + \sum_{n \mathop = 2}^\infty \frac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n }\) $a b$ cancels
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 1}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 1} } - \frac {a \paren {a + 1} b \paren {b + 1} } { c \paren {c + 1} } \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\) $x$ term cancels
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \dfrac {a b} c \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n }\) $x$ term cancels
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \frac {a b} { \, c} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 1}^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^{n + 1} }\) $x$ term cancels
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 2}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} x^n }\) $x$ term cancels
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \, n!} x^{n + 1} } - \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\) divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 2}^{\overline {n - 1} } \paren {b + 2}^{\overline {n - 1} } } {\paren {c + 2}^{\overline {n - 1} } n!} x^n }\) divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {a + 2}^{\overline {n - 1} } \paren {b + 2}^{\overline {n - 1} } } {\paren {c + 2}^{\overline {n - 1} } n!} x^{n + 1} }\) divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 2}^\infty \frac {\paren {a + 2}^{\overline {n - 2} } \paren {b + 2}^{\overline {n - 2} } } {\paren {c + 2}^{\overline {n - 2} } n!} x^n }\) divide through by $\dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } $
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 1}!} x^{n + 2} } - \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\) reindex sum
\(\ds \) \(\) \(\, \ds + \, \) \(\ds c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 2}!} x^{n + 2} }\) reindex sum
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} x^{n + 2} }\) reindex sum
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }\) reindex sum
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \dfrac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 1}!} x^{n + 2} } + c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline {n + 1} } \paren {b + 2}^{\overline {n + 1} } } {\paren {c + 2}^{\overline {n + 1} } \paren {n + 2}!} x^{n + 2} }\) rearranging
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} x^{n + 2} }\) rearranging
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} x^{n + 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} \frac {\paren {n + 2} } {\paren {n + 2} } \frac {\paren {a + 2 + n} \paren {b + 2 + n} } {\paren {c + 2 + n} } x^{n + 2} } + c \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \frac {\paren {a + 2 + n} \paren {b + 2 + n} } {\paren {c + 2 + n} } x^{n + 2} }\) Sum of Indices of Rising Factorial and multiplying top and bottom by $\paren {n + 2}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} n!} \frac {\paren {n + 1} \paren {n + 2} } {\paren {n + 1} \paren {n + 2} } x^{n + 2} }\) Sum of Indices of Rising Factorial and multiplying top and bottom by $\paren {n + 1} \paren {n + 2}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {a + b + 1} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 1}!} \frac {\paren {n + 2} } {\paren {n + 2} } x^{n + 2} }\) multiplying top and bottom by $\paren {n + 2}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \frac {\paren {a + 2 + n} \paren {b + 2 + n} \paren {n + 2 + c} } {\paren {c + 2 + n} } x^{n + 2} }\) Linear Combination of Convergent Series and Definition of Factorial
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!} \paren {n + 1} \paren {n + 2} x^{n + 2} }\) Definition of Factorial
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {n + 2} \paren {a + b + 1} x^{n + 2} }\) Definition of Factorial
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {a + 2 + n} \paren {b + 2 + n} x^{n + 2} }\) $\paren {n + 2 + c}$ cancels
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!} \paren {n + 1} \paren {n + 2} x^{n + 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {n + 2} \paren {a + b + 1} x^{n + 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds a b \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} x^{n + 2} }\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n } \paren {b + 2}^{\overline n } } {\paren {c + 2}^{\overline n } \paren {n + 2}!} \paren {a b + 2 a + n a + 2 b + 4 + 2 n + n b + 2 n + n^2 } x^{n + 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {a + 2}^{\overline n} \paren {b + 2}^{\overline n} } {\paren {c + 2}^{\overline n} \paren {n + 2}!} \paren {n^2 + 3 n + 2 + n a + n b + n + 2 a + 2 b + 2 + a b} x^{n + 2} }\)

$\blacksquare$


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Sources

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