NAME: Wolfram Koepf

 ADDRESS: Prof. Dr. Wolfram Koepf
 HTWK Leipzig
 Fachbereich IMN
 Postfach 30 00 66

 EMAIL ADDRESS: koepf@imn.htwk-leipzig.de

 TITLE OF TALK: A Maple Package on q-Hypergeometric Summation

 ABSTRACT OF TALK: is attached as LaTeX file (koepf_abstract.tex)
Basic or {\sl $q$-hypergeometric series} are power series 
$\sum\limits_{k=0}^\infty A_k\,x^k$ whose coefficients $A_k$ have
a term ratio $A_{k+1}/A_k\in\Q(q^k)$ which is rational w.r.t.\ $q^k$.
Such coefficients $A_k$ are called {\sl $q$-hypergeometric terms}.

The $q$-analog of the celebrated {\sl Gosper algorithm} decides whether a
$q$-hyper\-geometric term $a_k$ has a $q$-hypergeometric term 
{\sl antidifference}
$s_k$, i.e.
\[
a_{k}=s_{k+1}-s_k
\]
and computes it in the affirmative case. In this case, summation is trivial
\[
\sum_{k=u}^v a_k= a_{v+1}-a_u
\;.
\]
The $q$-analog of {\sl Zeilberger's algorithm} finds a recurrence equation 
for a $q$-hypergeometric series
\[
S(n)=\sum_{k\in\Z} F(n,k)
\]
s.th.\ $F(n,k)$ is a $q$-hypergeometric term with respect to both $n$ and $k$.
This recurrence equation is {\sl $q$-holonomic}, i.e., it is linear, 
homogeneous, and has polynomial coefficients w.r.t.\ $q^n$.

Finally, the $q$-analog of {\sl Petkovesek's algorithm} finds 
$q$-hypergeometric term solutions of $q$-holonomic recurrence equations.

The theory of hypergeometric and $q$-hypergeometric series as well as the
above algorithms are developed in \cite{2}.

In this talk, I present a Maple package, jointly developed with Harald B\"oing
\cite{1}, which deals with these three and some more algorithms.
Many applications, in particular to orthogonal polynomials and 
special functions, are given.

\begin{thebibliography}{9}
\bibitem{1}
B\"oing, Harald and Koepf, Wolfram:
Algorithms for $q$-hypergeometric Summation in Computer Algebra.
Konrad-Zuse-Zentrum Berlin (ZIB), Preprint SC 98--02, 1998,
to appear in: Journal Symbolic Computation, 1999.

\bibitem{2}
Koepf, Wolfram: {\sl Hypergeometric Summation.
An Algorithmic Approach to Summation and Special Function Identities.}
Vieweg, Braunschweig/Wiesbaden, 1998, \mbox{ISBN 3-528-06950-3}.

\end{thebibliography}



 
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