Definition:Hypergeometric Series

From ProofWiki
Jump to navigation Jump to search


A hypergeometric series is a power series:

$\ds \beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{k \mathop = 0}^\infty \beta_k z^k$

where $\beta_0 = 1$ and the ratio of successive coefficients is a rational function of $k$:

$\dfrac {\beta_{k + 1} } {\beta_k} = \dfrac {\map A k} {\map B k}$

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

Also see

The functions generated by hypergeometric series are called generalized hypergeometric functions:

$\ds \map { {}_m \operatorname F_n} { { {a_1, \ldots, a_m} \atop {b_1, \ldots, b_n} } \, \middle \vert \, z} = \sum_{k \mathop = 0}^\infty \dfrac { {a_1}^{\overline k} \cdots {a_m}^{\overline k} } { {b_1}^{\overline k} \cdots {b_n}^{\overline k} } \dfrac {z^k} {k!} = \sum_{k \mathop = 0}^\infty \beta_k z^k$

When $m = 2$ and $n = 1$, the function is referred to as a Gaussian hypergeometric function and $\beta_k$ is defined as:

$\ds \beta_k = \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} k!} $

where $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.

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)