Definition:Hypergeometric Series

Definition

A hypergeometric series is a power series:

\(\ds \map F {a, b, c; n}\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \alpha_n z^n\)

\(\ds \)

\(=\)

\(\ds \alpha_0 + \alpha_1 z + \alpha_2 z^2 + \cdots\)

where:

$\map F {a, b, c; n}$ denotes the Gaussian hypergeometric function

$\alpha_n = \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!}$

$a^{\overline n}$ denotes the $n$th rising factorial of $a$.

Also see

• Definition:Generalized Hypergeometric Function
• Definition:Hypergeometric Differential Equation

• Results about hypergeometric series can be found here.

Historical Note

Carl Friedrich Gauss did considerable work on this series, as published in his work:

1813: Disquisitiones generales circa seriam infinitam   $1 + \frac {\alpha \beta} {1 \cdot \gamma} \, x + \frac {\alpha \left({\alpha + 1}\right) \beta \left({\beta + 1}\right)} {1 \cdot 2 \cdot \gamma \left({\gamma + 1}\right)} \, x \, x + \cdots $ (Commentationes societatis regiae scientarum Gottingensis recentiores Vol. 2)

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypergeometric series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hypergeometric series

• Weisstein, Eric W. "Hypergeometric Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricSeries.html