Category:Definitions/Hypergeometric Functions

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This category contains definitions related to Hypergeometric Functions.
Related results can be found in Category: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 $x$.