Arcsine Function in terms of Gaussian Hypergeometric Function

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Theorem

$\arcsin x = x \, {}_2 \map {F_1} {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$

where:

$x$ is a real number with $\size x \le 1$
$\arcsin$ denotes the arcsine function
${}_2 F_1$ denotes the Gaussian hypergeometric function.


Proof

\(\ds x \, {}_2 \map {F_1} {\frac 1 2, \frac 1 2; \frac 3 2; x^2}\) \(=\) \(\ds x \sum_{n \mathop = 0}^\infty \frac {\paren {\paren {\frac 1 2}^{\bar n} }^2} {\paren {\frac 3 2}^{\bar n} } \frac {x^{2 n} } {n!}\) Definition of Gaussian Hypergeometric Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\map \Gamma {\frac 1 2 + n} }^2 \map \Gamma {\frac 3 2} } {\paren {\map \Gamma {\frac 1 2} }^2 \map \Gamma {\frac 3 2 + n} } \frac {x^{2 n + 1} } {n!}\) Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\map \Gamma {\frac 1 2 + n} }^2 \map \Gamma {\frac 1 2} } {2 \paren {\map \Gamma {\frac 1 2} }^2 \paren {\frac 1 2 + n} \map \Gamma {\frac 1 2 + n} } \frac {x^{2 n + 1} } {n!}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\map \Gamma {\frac 1 2 + n} } {\paren {2 n + 1} \sqrt \pi} \frac {x^{2 n + 1} } {n!}\) Gamma Function of One Half
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\paren {\frac {\paren {2 n}!} {2^{2 n} n!} \sqrt \pi} \times \frac 1 {\paren {2 n + 1} \sqrt \pi} \times \frac 1 {n!} x^{2 n + 1} }\) Gamma Function of Positive Half-Integer
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}\)
\(\ds \) \(=\) \(\ds \arcsin x\) Power Series Expansion for Real Arcsine Function

$\blacksquare$


Sources