A Telescoping Method for Double Summations
William Y.C. Chen, Qing-Hu Hou and Yan-Ping Mu
Abstract: We present a method to prove hypergeometric double summation identities. Given a hypergeometric term F(n, i, j), we aim to find a difference operator L = a_{0}(n)N ^{0} + a_{1}(n)N ^{1} + ··· + a_{r}(n)N^{ r} and rational functions R_{1}(n, i, j), R_{2}(n, i, j) such that LF = _{i}(R_{1}F) + _{j}(R_{2}F). Based on simple divisibility considerations, we show that the denominators of R_{1} and R_{2} must possess certain factors which can be computed from F(n, i, j). Using these factors as estimates, we may find the numerators of R_{1} and R_{2} by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, the Carlitz's identities, the Apéry- Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkovšek-Wilf-Zeilberger identity. AMS Classification: 33F10, 68W30 Keywords: Zeilberger's algorithm, double summation, hypergeometric term Download: pdf |