Definition:Hypergeometric Series

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A hypergeometric series is a power series:

$\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{n \mathop \ge 0} \beta_n z^n$

in which the ratio of successive coefficients is a rational function of $n$:

$\dfrac {\beta_{n + 1} } {\beta_n} = \dfrac {\map A n} {\map B n}$

where $\map A n$ and $\map B n$ are polynomials in $n$.

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)