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\chapter{SPECFN2: Special special functions}
\label{SPECFN2}
\typeout{{SPECFN2: Package for special special functions}}

{\footnotesize
\begin{center}
Victor S. Adamchik \\
Byelorussian University \\
Minsk, Belorus \\[0.1in]
and\\[0.05in]
Winfried Neun \\
Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\
Takustra\"se 7 \\
D--14195 Berlin--Dahlem, Germany \\[0.05in]
e--mail: neun@zib.de

\end{center}
}
\ttindex{SPECFN2}
\index{Generalised Hypergeometric functions}
\index{Meijer's G function}

The (generalised) hypergeometric functions  

\begin{displaymath}
_pF_q \left( {{a_1, \ldots , a_p} \atop {b_1, \ldots ,b_q}} \Bigg\vert z \right)
\end{displaymath}

are defined in textbooks on special functions.
\section{\REDUCE{} operator HYPERGEOMETRIC}

The operator {\tt hypergeometric} expects 3 arguments, namely the 
list of upper parameters (which may be empty), the list of lower
parameters (which may be empty too), and the argument, e.g:

\begin{verbatim}

hypergeometric ({},{},z);

 Z
E

hypergeometric ({1/2,1},{3/2},-x^2);

 ATAN(X)
---------
    X
\end{verbatim}


\section{Enlarging the HYPERGEOMETRIC operator}

Since hundreds of particular cases for the generalised hypergeometric
functions can be found in the literature, one cannot expect that all
cases are known to the {\tt hypergeometric} operator.
Nevertheless the set of special cases can be augmented by adding
rules to the \REDUCE{} system, {\em e.g.}
\begin{verbatim}
let {hypergeometric({1/2,1/2},{3/2},-(~x)^2) => asinh(x)/x};
\end{verbatim}