Power of One plus x in terms of Gaussian Hypergeometric Function

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Theorem

$\map F {-p, 1; 1; -x} = \paren {1 + x}^p$

where:

$x$ and $p$ are real numbers with $\size x < 1$
$F$ denotes the Gaussian hypergeometric function.


Proof

\(\ds \map F {-p, 1; 1; -x}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {-x}^n} {n!}\) Definition of Gaussian Hypergeometric Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\prod_{j \mathop = 0}^{n - 1} \paren {-p + j} } \frac {x^n} {n!}\) Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\prod_{j \mathop = 0}^{n - 1} \paren {-1} } \paren {\prod_{j \mathop = 0}^{n - 1} \paren {-p + j} } \frac {x^n} {n!}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\prod_{j \mathop = 0}^{n - 1} \paren {p - j} } \frac {x^n} {n!}\) Product of Products
\(\ds \) \(=\) \(\ds \paren {1 + x}^p\) General Binomial Theorem

$\blacksquare$


Sources

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