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An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities. (English) Zbl 0739.05007
It is shown that every ‘proper-hypergeometric’ multisum/integral identity, or $q$-identity, with fixed number of summation and/or integration signs, possesses a short, computer-constructible proof. We give a fast algorithm for finding such proofs. Most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here. We give many examples of the method, including computer-generated proofs of identities of Mehta-Dyson, Selberg, Hille-Hardy, $q$-Saalschütz, and others. The prospect of using the method for proving multivariate identities that involve an arbitrary number of summations/integrations is discussed.
Reviewer: H.S.Wilf

##### MSC:
 05A19 Combinatorial identities, bijective combinatorics 05A10 Combinatorial functions 11B65 Binomial coefficients, etc. 05A30 $q$-calculus and related topics 33C99 Hypergeometric functions 39A10 Additive difference equations