# Power of One plus x in terms of Gaussian Hypergeometric Function

## Theorem

$\ds {}_2 \map {F_1} {-p, 1; 1; -x} = \paren {1 + x}^p$

where:

$x$ and $p$ are real numbers with $\size x < 1$
${}_2 F_1$ denotes the Gaussian hypergeometric function.

## Proof

 $\ds {}_2 \map {F_1} {-p, 1; 1; -x}$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {-x}^n} {n!}$ 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

$\blacksquare$