Since the 1990's, Zeilberger's method of creative
telescoping has played an important role in the automatic verification of
special-function identities. The long-term goal initiated in this work is to
obtain fast algorithms and implementations for definite integration and
summation in the framework of this method. Our contributions include new
practical algorithms, complexity analyses of algorithms, and theoretical
criteria for the termination of existing algorithms. On the practical side,
we present a new algorithm for computing minimal telescopers for bivariate
rational functions. This algorithm is based on Hermite reduction. We also
improve the classical Almkvist and Zeilberger's algorithm for
rational-function inputs. The Hermite-reduction based algorithm and improved
Almkvist and Zeilberger's algorithm are analyzed in terms of field
operations. Both complexity analysis and experimental results show that our
algorithms are superior to other known ones for rational-function inputs. On
the theoretic side, we present a structure theorem for multivariate
hyperexponential-hypergeometric functions. This theorem is based on
(multivariate) Christoper's theorem for hyperexponential functions, the
Ore-Sato theorem for hypergeometric terms, and our generalization of a recent
result by Feng, Singer, and Wu on compatible bivariate continuous-discrete
rational functions. The structure theorem allows us to decomposes a
hyperexponential-hypergeometric function as a product of a rational function,
several exponential and power functions, and factorial terms. Furthermore, we
derive two criteria for the existence of telescopers for bivariate
hyperexponential-hypergeometric functions: one is with respect to the
continuous variable, and the other with respect to the discrete one. The two
criteria solve the termination problems of the continuous-discrete analogue
of Zeilberger's algorithms.
