Kummer's Hypergeometric Theorem/Examples/2F1(0.4,0.1;1.3;-1)

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Example of Use of Kummer's Hypergeometric Theorem

$1 - \paren {\dfrac {\paren {2} \paren {1} } {\paren {5} \paren {13} } } + \paren {\dfrac {\paren {2 \times 7} \paren {1 \times 11} } {\paren {5 \times 10} \paren {13 \times 23} } } - \paren {\dfrac {\paren {2 \times 7 \times 12} \paren {1 \times 11 \times 21} } {\paren {5 \times 10 \times 15} \paren {13 \times 23 \times 33} } } + \cdots = \dfrac {1944^{\frac 1 5} \pi^{\frac 3 2} } {\phi \map \Gamma {\dfrac 1 {10} } \paren {\map \Gamma {\dfrac 7 {10} } }^2 }$


Proof

From Kummer's Hypergeometric Theorem:

$\ds \map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} } $

where:

$\ds \map F {n, -x; x + n + 1; -1}$ is the Gaussian hypergeometric function of $-1$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


We have:

\(\ds \map F {\dfrac 2 5, \dfrac 1 {10}; \dfrac {13} {10}; -1}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac 2 5}^{\overline k} \paren {\dfrac 1 {10} }^{\overline k} } { \paren {\dfrac {13} {10} }^{\overline k} } \dfrac {\paren {-1}^k} {k!}\) Definition of Gaussian Hypergeometric Function
\(\ds \) \(=\) \(\ds 1 - \dfrac {\paren {\dfrac 2 5} \paren {\dfrac 1 {10} } } {\paren {\dfrac {13} {10} } 1!} + \dfrac {\paren {\dfrac 2 5 \times \dfrac 7 5} \paren {\dfrac 1 {10} \times \dfrac {11} {10} } } {\paren {\dfrac {13} {10} \times \dfrac {23} {10} } 2!} - \dfrac {\paren {\dfrac 2 5 \times \dfrac 7 5 \times \dfrac {12} 5} \paren {\dfrac 1 {10} \times \dfrac {11} {10} \times \dfrac {21} {10} } } {\paren {\dfrac {13} {10} \times \dfrac {23} {10} \times \dfrac {33} {10} } 3!} + \cdots\) One to Integer Rising is Integer Factorial, $1^k = 1$, Number to Power of Zero Rising is One
\(\ds \) \(=\) \(\ds 1 - \paren {\dfrac {\paren {2} \paren {1} } {\paren {5} \paren {13} } } + \paren {\dfrac {\paren {2 \times 7} \paren {1 \times 11} } {\paren {5 \times 10} \paren {13 \times 23} } } - \paren {\dfrac {\paren {2 \times 7 \times 12} \paren {1 \times 11 \times 21} } {\paren {5 \times 10 \times 15} \paren {13 \times 23 \times 33} } } + \cdots\)

and:

\(\ds \map F {\dfrac 2 5, \dfrac 1 {10}; \dfrac {13} {10}; -1}\) \(=\) \(\ds \dfrac {\map \Gamma {-\dfrac 1 {10} + \dfrac 2 5 + 1} \map \Gamma {\dfrac {\dfrac 2 5} 2 + 1} } {\map \Gamma {-\dfrac 1 {10} + \dfrac {\dfrac 2 5} 2 + 1} \map \Gamma {\dfrac 2 5 + 1} }\) Kummer's Hypergeometric Theorem
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {\dfrac {13} {10} } \map \Gamma {\dfrac 6 5} } {\map \Gamma {\dfrac {11} {10} } \map \Gamma {\dfrac 7 5} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\dfrac 3 {10} \map \Gamma {\dfrac 3 {10} } \dfrac 1 5 \map \Gamma {\dfrac 1 5 } } {\dfrac 1 {10} \map \Gamma {\dfrac 1 {10} } \dfrac 2 5 \map \Gamma {\dfrac 2 5} }\) Definition of Gamma Function
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac {3 \map \Gamma {\dfrac 3 {10} } \map \Gamma {\dfrac 1 5 } } {2 \map \Gamma {\dfrac 1 {10} } \map \Gamma {\dfrac 2 5} }\)


Recall from the Euler Reflection Formula: $\map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

Therefore:

\(\ds \map \Gamma {\dfrac 3 {10} } \map \Gamma {1 - \dfrac 3 {10} }\) \(=\) \(\ds \dfrac \pi {\map \sin {\dfrac {3\pi} {10} } }\)
\(\ds \map \Gamma {\dfrac 3 {10} } \map \Gamma {\dfrac 7 {10} }\) \(=\) \(\ds \dfrac \pi {\dfrac \phi 2 }\) Sine of Complement: $\map \sin {\dfrac \pi 2 - \dfrac \pi 5} = \cos \dfrac \pi 5$ and Golden Ratio: $\cos \dfrac \pi 5 = \dfrac \phi 2$
\(\text {(2)}: \quad\) \(\ds \map \Gamma {\dfrac 3 {10} }\) \(=\) \(\ds \dfrac {2 \pi} {\phi \map \Gamma {\dfrac 7 {10} } }\)

Recall from the Legendre Duplication Formula: $\map \Gamma z \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$

Therefore:

\(\ds \map \Gamma {\dfrac 1 5} \map \Gamma {\dfrac 1 5 + \dfrac 1 2}\) \(=\) \(\ds 2^{1 - \frac 2 5} \sqrt \pi \, \map \Gamma {\dfrac 2 5}\)
\(\text {(3)}: \quad\) \(\ds \dfrac {\map \Gamma {\dfrac 1 5} } {\map \Gamma {\dfrac 2 5} }\) \(=\) \(\ds \dfrac {2^{1 - \frac 2 5} \sqrt \pi} {\map \Gamma {\dfrac 7 {10} } }\)


Substituting these results back into our equation above:

\(\ds \map F {\dfrac 2 5, \dfrac 1 {10}; \dfrac {13} {10}; -1}\) \(=\) \(\ds \dfrac {3 \map \Gamma {\dfrac 3 {10} } \map \Gamma {\dfrac 1 5 } } {2 \map \Gamma {\dfrac 1 {10} } \map \Gamma {\dfrac 2 5} }\) from $\paren {1}$ above
\(\ds \) \(=\) \(\ds \dfrac {3 \paren {\dfrac {2 \pi} {\phi \map \Gamma {\dfrac 7 {10} } } } } {2 \map \Gamma {\dfrac 1 {10} } } \times \dfrac {2^{1 - \frac 2 5} \sqrt \pi} {\map \Gamma {\dfrac 7 {10} } }\) substituting $\paren {2}$ and $\paren {3}$ into $\paren {1}$
\(\ds \) \(=\) \(\ds \dfrac {1944^{\frac 1 5} \pi^{\frac 3 2} } {\phi \map \Gamma {\dfrac 1 {10} } \paren {\map \Gamma {\dfrac 7 {10} } }^2}\) $1944 = 3^5 \times 2^3$


Therefore:

$1 - \paren {\dfrac {\paren {2} \paren {1} } {\paren {5} \paren {13} } } + \paren {\dfrac {\paren {2 \times 7} \paren {1 \times 11} } {\paren {5 \times 10} \paren {13 \times 23} } } - \paren {\dfrac {\paren {2 \times 7 \times 12} \paren {1 \times 11 \times 21} } {\paren {5 \times 10 \times 15} \paren {13 \times 23 \times 33} } } + \cdots = \dfrac {1944^{\frac 1 5} \pi^{\frac 3 2} } {\phi \map \Gamma {\dfrac 1 {10} } \paren {\map \Gamma {\dfrac 7 {10} } }^2 }$

$\blacksquare$


Sources

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