Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities
The current book is the result of a lecture course that I gave at the Free Univer sity, Berlin, during the spring semester 1995. This course was influenced by the remarkable book Concrete Matbematics by Graham, Knuth and Patashnik, and by the interesting lecture notes Identities and Tbeir Computer Proofs by Herbert Wilf [Wilf93]. In the meantime these notes appeared tagether with other material in the book A = B by Petkovsek, Wilf and Zeilberger [PWZ96]. In cantrast to the books just mentioned, it is my objective to present the material by giving more detailed advice on implementation. Furthermore I wished to cover not only material about recurrence equations but also about differential equations, not only about sums but also about integrals, and finally not only the hypergeometric case but also its q-analogue. In the current book, up-to-date algorithmic techniques for summation are described in detail, and worked out using Maple programs. With Maple release V.4 and higher, some of these tools are available through Maple's sum command and sumtools pack age, by an implementation that I incorporated in the Maple library prior to my lecture course. In this book, readers are invited to implement the algorithms step by step. This will give them a detailed insight in the structure of the algorithms under consideration, and will enable them to solve quite involved problems.
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1998 Wolfram Koepf Algorithm 10.2 Algorithm 2.1 antiderivative application of Gosper's applied binomial coefficients binomial theorem calculation chapter Chu-Vandermonde identity computation computer algebra system Copyright 1998 Wolfram database deduce degree bound denominator denote equate coefficients equation of order ERROR('algorithm Example Exercise finite support formula g-analogues given Gosper-summable Gosper's algorithm Hence holonomic recurrence equation hyperexponential term Hypergeom hypergeometric function hypergeometric identities hypergeometric representation hypergeometric summation hypergeometric term solutions implementation initial values input integer integer-linear integral Jacobi polynomials Konrad-Zuse-Zentrum Berlin Laguerre polynomials Legendre polynomials Lemma linear system lower parameters Maple procedure nonnegative integer Note option Copyright 1998 order recurrence equation orthogonal polynomials Petkovsek's algorithm Pn(x Pochhammer symbols proof rational certificate rational functions recurrence and differential rewriting right-hand side Rodrigues formula satisfies Session Show sn+i summand term ratio term with respect Theorem valid WZ method Zeilberger Zeilberger's algorithm zero