Definition:Hypergeometric Function/Generalized

From ProofWiki
Jump to navigation Jump to search

Definition


This article, or a section of it, needs explaining.
In particular: Specify the domain of the numbers involved: $a_1$ to $a_m$, $b_1$ to $b_m$, $z$ -- complex is assumed, but this need to be made certain of
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.



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:

$\ds \map { {}_2 F_1} {a, b; c; z} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {z^k} {k!}$


  • Results about hypergeometric functions can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Hypergeometric_Function/Generalized&oldid=687545"