Gauss's Hypergeometric Theorem/Examples/2F1(1,1;2.5;1)
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Example of Use of Gauss's Hypergeometric Theorem
- $1 + \dfrac 2 5 + \paren {\dfrac {2 \times 4} {5 \times 7} } + \paren {\dfrac {2 \times 4 \times 6} {5 \times 7 \times 9} } + \cdots = 3$
Proof
From Gauss's Hypergeometric Theorem:
- $\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$
where:
- $\map F {a, b; c; 1}$ is the Gaussian hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
- $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
- $\map \Gamma {n + 1} = n!$ is the Gamma function.
We have:
| \(\ds \map F {1, 1; \dfrac 5 2; 1}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac { 1^{\overline k} 1^{\overline k} } { \paren {\dfrac 5 2}^{\overline k} } \dfrac {1^k} {k!}\) | Definition of Gaussian Hypergeometric Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac { 1^{\overline k} } { \paren {\dfrac 5 2}^{\overline k} }\) | One to Integer Rising is Integer Factorial, $1^k = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \dfrac 1 {\paren {\dfrac 5 2} } + \paren {\dfrac {1 \times 2} {\dfrac 5 2 \times \dfrac 7 2} } + \paren {\dfrac {1 \times 2 \times 3} {\dfrac 5 2 \times \dfrac 7 2 \times \dfrac 9 2} } + \cdots\) | Number to Power of Zero Rising is One | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \dfrac 2 5 + \paren {\dfrac {2 \times 4} {5 \times 7} } + \paren {\dfrac {2 \times 4 \times 6} {5 \times 7 \times 9} } + \cdots\) | rearranging |
and:
| \(\ds \map F {1, 1; \dfrac 5 2; 1}\) | \(=\) | \(\ds \dfrac {\map \Gamma {\dfrac 5 2} \map \Gamma {\dfrac 5 2 - 1 - 1} } {\map \Gamma {\dfrac 5 2 - 1} \map \Gamma {\dfrac 5 2 - 1} }\) | Gauss's Hypergeometric Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\map \Gamma {\dfrac 5 2} \map \Gamma {\dfrac 1 2} } {\map \Gamma {\dfrac 3 2} \map \Gamma {\dfrac 3 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\dfrac {3 \sqrt \pi} 4 \times \sqrt \pi} {\dfrac {\sqrt \pi} 2 \times \dfrac {\sqrt \pi} 2}\) | Gamma Function of $\dfrac 1 2$, Gamma Function of $\dfrac 3 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3\) |
Therefore:
- $1 + \dfrac 2 5 + \paren {\dfrac {2 \times 4} {5 \times 7} } + \paren {\dfrac {2 \times 4 \times 6} {5 \times 7 \times 9} } + \cdots = 3$
$\blacksquare$