Titlebook: Hypergeometric Summation; An Algorithmic Appro Wolfram Koepf Textbook 2014Latest edition Springer-Verlag London 2014 Algorithmic Summation.

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Hypergeometric Database,In this chapter we list some of the major hypergeometric identities. Note that most of these do not require any variables to have integer values. We give examples showing how this database can be used to generate binomial identities.

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The Wilf-Zeilberger Method,In this chapter, we study the connection between Gosper’s algorithm and definite sums.Firstly, we give a direct application of Gosper’s algorithm to definite summation. The application of Gosper’s algorithm to a modified input can prove definite hypergeometric identities.

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,Zeilberger’s Algorithm,In this chapter, we introduce Zeilberger’s extension of Gosper’s algorithm, using which one can not only prove hypergeometric identities but also sum definite series in many cases, if they represent hypergeometric terms.

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Extensions of the Algorithms,In this chapter, we extend Gosper’s, Wilf-Zeilberger’s and Zeilberger’s methods to accept rational-linear inputs rather than only integer-linear ones. For such an input . is not always rational, so that Gosper’s algorithm may not apply.

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Hyperexponential Antiderivatives,In this chapter, we consider a continuous counterpart of Gosper’s algorithm. The appropriate question is to find a hyperexponential term antiderivative .(.) of a given .(.) whenever one exists.

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Holonomic Equations for Integrals,Now we are ready to consider . of hyperexponential terms. If the corresponding indefinite integral is a hyperexponential term again, then the continuous Gosper algorithm applies, and definite integration is trivial.

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