Definition:Hypergeometric Series
Definition
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 $n$.
Needs to include a derivation of the notation ${}_p F_q \tuple {a_1, \ldots, a_p; b_1, \ldots, b_q; z}$
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The Penguin Dictionary of Mathematics, 4th ed. defines this where the coefficient $\beta_n$ is specifically $\dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!}$
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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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: hypergeometric series