Abstract:  We present an algorithm to prove hypergeometric double summa- tion identities. Given a hypergeometric term F(n, i, j), we aim to nd a dierence operator L = a₀(n)N⁰ + a₁(n)N¹ + ... + a_r(n)N^r and rational functions R₁(n, i, j), R₂(n, i, j) such that LF = _i(R₁F) +_j(R₂F). Based on simple divisibility considerations, we show that the denominators of R₁ and R₂ must possess certain factors which can be computed from F(n, i, j). Using these factors as estimates, we may nd the numerators of R₁ and R₂ by guessing the upper bounds of the degrees and solving systems of linear equations. Our algorithm is valid for the Andrews-Paule identity, the Carlitz's identi- ties, the Apéry-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkovsek-Wilf-Zeilberger identity.

  AMS Classification:  33F10, 68W30.

  Keywords:  Zeilberger's algorithm, double summation, hypergeometric term, Andrews-Paule identity.

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