Category:Hypergeometric Functions
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This category contains results about Hypergeometric Functions.
Definitions specific to this category can be found in Definitions/Hypergeometric Functions.
Gaussian Hypergeometric Function
The Gaussian hypergeometric function is an instance of a generalized hypergeometric function, given for $\size z < 1$ by:
\(\ds \map F {a, b; c; z}\) | \(:=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} } \dfrac {z^n} {n!}\) | where $x^{\overline n}$ denotes the $n$th rising factorial power of $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \dfrac {a b} {1! \, c} z + \dfrac {a \paren {a + 1} b \paren {b + 1} } {2! \, c \paren {c + 1} } z^2 + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! \, c \paren {c + 1} \paren {c + 2} } z^3 + \cdots\) |
Generalized Hypergeometric Function
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 $x$.
Subcategories
This category has the following 10 subcategories, out of 10 total.
D
G
H
- Hypergeometric Series (empty)
K
P
- Pfaff-Saalschütz Theorem (3 P)
T
- Thomae's Transformation (2 P)
W
Pages in category "Hypergeometric Functions"
The following 15 pages are in this category, out of 15 total.
D
- Dixon's Hypergeometric Theorem
- Dougall's Hypergeometric Theorem
- Dougall's Hypergeometric Theorem/Corollary 1
- Dougall's Hypergeometric Theorem/Corollary 2
- Dougall's Hypergeometric Theorem/Corollary 3
- Dougall's Hypergeometric Theorem/Corollary 4
- Dougall's Hypergeometric Theorem/Corollary 5
- Dougall's Hypergeometric Theorem/Corollary 6
- Dougall-Ramanujan Identity