Reciprocal of One minus x in terms of Gaussian Hypergeometric Function

From ProofWiki
Jump to navigation Jump to search

Theorem

$\dfrac 1 {1 - x} = {}_2 \map {F_1} {1, p; p; x}$

where:

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


Proof

\(\ds {}_2 \map {F_1} {1, p; p; x}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {1^{\bar n} p^{\bar n} } {p^{\bar n} } \frac {x^n} {n!}\) Definition of Gaussian Hypergeometric Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty n! \frac {x^n} {n!}\) One to Integer Rising is Integer Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty x^n\)
\(\ds \) \(=\) \(\ds \frac 1 {1 - x}\) Sum of Infinite Geometric Sequence, justified as $\size x < 1$

$\blacksquare$


Sources