Definition:Hypergeometric Function/Generalized
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Definition
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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
- 1984: Jacques Dutka: The Early History of the Hypergeometric Function (Arch. Hist. Exact Sci. pp. 15 – 34) www.jstor.org/stable/41133728
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(39)$
- Weisstein, Eric W. "Hypergeometric Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricFunction.html