Definition:Hypergeometric Function/Gaussian

Definition

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 $z$

\(\ds \)

\(=\)

\(\ds 1 + \dfrac {a b} {1! \, c} z + \dfrac {a \paren {a + 1} b \paren {b + 1} } {2! \, c \paren {c + 1} } z^2 + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! \, c \paren {c + 1} \paren {c + 2} } z^3 + \cdots\)

Also denoted as

Using the notation of the generalized hypergeometric function, the Gaussian hypergeometric function can be denoted as:

$\ds \map { {}_2 F_1} {a, b; c; z}$

or if necessary to draw particular attention to where it came from:

$\ds \map { {}_2 F_1} { { {a, b} \atop c} \, \middle \vert \, z}$

Also known as

The Gaussian hypergeometric function is also known as the ordinary hypergeometric function, or just the hypergeometric function

Some sources refer to this as Gauss's hypergeometric function or (grammatically inaccurately) Gauss' hypergeometric function.

Also see

• Solution to Hypergeometric Differential Equation

• Results about the Gaussian hypergeometric function can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.

Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $14.1$: The hypergeometric series
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Hypergeometric Functions: $31.2$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypergeometric function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hypergeometric function