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We observe that many special functions are solutions of so-called holonomic systems. Bernstein's deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps. This is partially substantiated by an algorithm that proves terminating hypergeometric series identities, and that is given both in English and in MAPLE.

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n) := 1!4!7!···(3n−2)! n!(n+1)!···(2n−1)! . Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) '1' of the first row is at the r th column, equals A(n) n+r−2 n−1 2n−1−r n−1

An algorithm for de6nite hypergeometric summation is given. It is based, in a non-obvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems.

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