Definition:Hypergeometric Function/Generalized

Definition
A generalized hypergeometric function is a function which can be defined in the form of a hypergeometric series defined as:


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

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

Also denoted as
Some sources denote this as:


 * $\ds \map { {}_m \operatorname F_n} {a_1, \ldots, a_m, b_1, \ldots, b_n; z}$

Also known as
Many sources refer to a generalized hypergeometric function merely as the hypergeometric function, considering the Gaussian hypergeometric function merely as an instance of this.

Also see
When $m = 2$ and $n = 1$, the generalized hypergeometric function reduces to a Gaussian hypergeometric function:
 * $\ds \map { {}_2 F_1} {a, b; c; z} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {z^k} {k!}$