The Hypergeometric Function

For more information on the Hypergeometric Series, see Husemöller [Husemöller, 1987], 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.