Reciprocal of One minus x in terms of Gaussian Hypergeometric Function
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Theorem
- $\dfrac 1 {1 - x} = \map F {1, p; p; x}$
where:
- $x$ and $p$ are real numbers with $\size x < 1$
- $F$ denotes the Gaussian hypergeometric function.
Proof
\(\ds \map F {1, p; p; x}\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {1^{\bar n} p^{\bar n} } {p^{\bar n} } \frac {x^n} {n!}\) | 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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Special Cases: $31.10$