Definition:Hypergeometric Series

Definition
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 $k$.

The functions generated by hypergeometric series are called generalized hypergeometric functions and are written as:


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

When $m = 2$ and $n = 1$, the function is simply referred to as a 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 $x$.