# The Hypergeometric Function

For more information on the Hypergeometric Series, see Husemöller [[Husemöller, 1987](../../references.md#cite-husem87)], page 176.

## `HypergeometricSeries(a, b, c, z): RngElt, RngElt, RngElt, RngElt -> RngElt`

Return the hypergeometric series $F(a,b,c;z)$ defined by

$$
F(a,b,c;z) = \sum_{0\le n} {{(a)_n(b)_n}\over{n!(c)_n} z^n}
$$

where $(a)_n = a (a+1) \cdots (a+n-1)$.

## `HypergeometricU(a, b, s): FldReElt, FldReElt, FldReElt -> FldReElt`

For positive real $s$ and complex arguments $a$ and $b$ this function returns the value of the confluent hypergeometric function $U(a, b, s)$. This can be defined by

$$
U(a, b, s)={1\over\Gamma(a)}\int_{u=0}^\infty e^{-su}u^{a-1}(1+u)^{b-a-1)}du.
$$

Pari is used here.